badge Datos personales

Pedro Morin

foto de Pedro Morin
  • email: morinpedro@gmail.com
  • DNI: 22070980
  • CUIL: 23-22070980-9
  • Estado civil: Casado
  • Domicilio: Los Claveles 6760. Colastiné Norte. 3001 Santa Fe. Argentina
  • Lugar y Fecha de Nacimiento: Santa Fe, 6 de julio de 1971.

school Estudios completados

lan Experiencia laboral

group Características personales

sticky_note_2 Publicaciones

  1. Direct estimates for adaptive time-stepping finite element methods, Journal of Complexity (2025) 101918.  Con Marcelo Actis, Fernando Gaspoz, Cornelia Schneider y Nick Schneider. 

    Abstract: We study direct estimates for adaptive time-stepping finite element methods for time-dependent partial differential equations. Our results generalize previous findings from \cite{AMS22}, where the approximation error was only measured in $L_2([0,T],L_2(\Omega))$. In particular, we now also cover the error norms $L_{\infty}([0,T],L_2(\Omega))$ and $L_2([0,T],H^1(\Omega))$ which are more natural in this context.

    Keywords: Direct estimates; Approximation classes; Besov spaces; Adaptive time-stepping finite element methods; Near-best approximation

    AMS Subject Classifications: primary: 41A25; 65D05. secondary: 65N30; 65N50.

    Artículo completo

  2. Error analysis for local coarsening in univariate spline spaces, Applied Mathematics and Computation (2024) 128616.  Con Silvano Figueroa y Eduardo M. Garau. 

    Abstract: In this article we analyze the error produced by the removal of an arbitrary knot from a spline function; we consider the $L^2$-, the $H^1$- and the $L^\infty$-errors. When a knot has multiplicity greater than one, this implies a reduction of its multiplicity by one unit. In particular, we deduce a very simple formula to compute the error in terms of some neighboring knots and a few coefficients of the considered spline. Furthermore, we show precisely how this error is related to the jump of a derivative of the spline at the knot. We then use the developed theory to propose efficient and very low-cost local error indicators and adaptive coarsening algorithms. Finally, we present some numerical experiments to illustrate their performance and show some applications.

    Artículo completo

  3. Approximation classes for adaptive time-stepping finite element methods, IMA Journal of Numerical Analysis (2022) 2817—2855.  Con Marcelo Actis y Cornelia Schneider. 

    Abstract: We study approximation classes for adaptive time-stepping finite element methods for time-dependent partial differential equations. We measure the approximation error in $L_2([0,T)\times \Omega)$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a by-product we define anisotropic Besov spaces for Banach-space-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.

    Keywords: approximation classes, Besov spaces, Whitney estimate, adaptive time-stepping finite element methods, greedy algorithm

    Artículo completo

  4. Convective transport in nanofluids: regularity of solutions and error estimates for finite element approximations, Journal of Mathematical Fluid Mechanics 23 (2021) 1—17.   Con Eberhard Bänsch. 

    Abstract: We study the stationary version of a thermodynamically consistent variant of the Buongiorno model describing convective transport in nanofluids. Under some smallness assumptions it is proved that there exist regular solutions. Based on this regularity result, error estimates, both in the natural norm as well as in weaker norms for finite element approximations can be shown. The proofs are based on the theory developed by Caloz and Rappaz for general nonlinear, smooth problems. Computational results confirm the theoretical findings.

    Artículo completo

  5. Convective transport in nanofluids: the stationary problem. Journal of Mathematical Analysis and Applications (2020) 124151.   Con Eberhard Bänsch y Sara Faghih-Naini. 

    Abstract: We analyze the existence of solutions to the stationary problem from a mathematical model for convective transport in nanofluids including thermophoretic effects that is very similar to the celebrated model of Buongiorno [6].

    Keywords: NanofluidThermophoresisHeat transferEnergy estimateWeak solution

    Artículo completo

  6. A new perspective on hierarchical spline spaces for adaptivity. Computers and Mathematics with Applications (2020) 2276—2303.   Con Marcelo Actis y M. Sebastián Pauletti. 

    Abstract: We introduce a framework for spline spaces of hierarchical type, based on a parent–children relation, which is very convenient for the analysis as well as the implementation of adaptive isogeometric methods. This framework exploits the innate refinement by functions in the B-splines context, rather than by elements or cells, which is more natural in the finite element context. Furthermore, it entails a new language to handle hierarchical spline spaces, which allows to tackle fundamental questions in a very simple manner. For example, it makes it simple to create hierarchical bases with several desired properties with a refinement procedure which has linear complexity, i.e., the resulting bases have cardinality bounded by the number of initially marked functions.

    Keywords: Hierarchical spacesSplines basesOptimality for adaptive methods

    Artículo completo

  7. The shape derivative of the Gauss curvature. Revista de la Unión Matemática Argentina (2018) 311—337.  Con Aníbal Chicco-Ruiz y M. Sebastián Pauletti. 

    Abstract: We present a review of results about the shape derivatives of scalar- and vector-valued shape functions, and extend the results from Do˘gan and Nochetto [ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 59—79] to more general surface energies. In that article, Do˘gan and Nochetto consider surface energies defined as integrals over surfaces of functions that can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature (a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of any finite dimension. As an application of the results, with relevance for numerical methods in applied problems, we derive a Newton-type method to approximate a minimizer of a shape functional. We finally find the particular formulas for the first and second order shape derivatives of the area and the Willmore functional, which are necessary for the aforementioned Newton-type method.

    Keywords: Shape derivative, Gauss curvature, shape optimization, differentiation formulas.

    AMS Subject Classifications: 65K10, 49M15, 53A10, 53A55

    Artículo completo

  8. Adaptive control of local errors for elliptic problems using weighted Sobolev norms. Numerical Methods for Partial Differential Equations 33 (2017) 1266—1282.  Con Eduardo M. Garau. 

    Abstract: We develop an a posteriori error estimator which focuses on the local H1 error on a region of interest. The estimator bounds a weighted Sobolev norm of the error and is efficient up to oscillation terms. The new idea is very simple and applies to a large class of problems. An adaptive method guided by this estimator is implemented and compared to other local estimators, showing an excellent performance.

    Artículo completo

  9. High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates, Foundations of Computational Mathematics 16 (2016) 1473—1539.  Con Andrea Bonito, J. Manuel Cascón, Khamron Mekchay y Ricardo H. Nochetto. 

    Abstract: We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W_\infty^1 and piecewise in a suitable Besov class embedded in C^{1,\alpha} with $\alpha\in(0,1]$. The idea is to have the surface sufficiently well resolved in relative to the current resolution of the PDE in H^1. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in and PDE error in .

    Keywords:

    AMS Subject Classifications:

    Artículo completo

  10. An algorithm for prescribed mean curvature using isogeometric methods Journal of Computational Physics 317 (2016) 185—203.  Con Aníbal Chicco-Ruiz y M. Sebastián Paulett. 

    Abstract: We present a Newton type algorithm to find parametric surfaces of prescribed mean curvature with a fixed given boundary. In particular, it applies to the problem of minimal surfaces. The algorithm relies on some global regularity of the spaces where it is posed, which is naturally fitted for discretization with isogeometric type of spaces. We introduce a discretization of the continuous algorithm and present a simple implementation using the recently released isogeometric software library igatools. Finally, we show several numerical experiments which highlight the convergence properties of the scheme.

    Keywords: Minimal surfaces; Prescribed curvature; Isogeometric analysis; Quasi-Newton

    Artículo completo

  11. A posteriori error estimates with point sources in fractional sobolev spaces, Numerical Methods for Partial Differential Equations (2016).  Con Fernando Gaspoz y Andreas Veeser. 

    Abstract: We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a~posteriori estimators with a specifically tailored oscillation and show that, on two-dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decayrates.

    Keywords: finite element methods, a~posteriori error estimators, Dirac mass, adaptivity, fractional Sobolev spaces

    Artículo completo

  12. Nonlocal diffusions on fractals: qualitative properties and numerical approximations, IMA Journal of Numerical Analysis (2016) 1143—1166.  Con Marilina Carena y Marcelo Actis. 

    Abstract: We propose a numerical method to approximate the solution of a nonlocal diffusion problem on a general setting of metric measure spaces. These spaces include, but are not limited to, fractals, manifolds and Euclidean domains. We obtain error estimates in $L^\infty(L^p)$ for $p=1,\infty$ under the sole assumption of the initial datum being in $L^p$. An improved bound for the error in $L^\infty(L^1)$ is obtained when the initial datum is in $L^2$. We also derive some qualitative properties of the solutions like stability, comparison principles and study the asymptotic behavior as $t\to\infty$. We finally present two examples on fractals: the Sierpinski gasket and the Sierpinski carpet, which illustrate on the effect of nonlocal diffusion for piecewise constant initialdatum.

    Keywords: non-local diffusions, space of homogeneous type, fractals

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  13. On a dissolution–diffusion model. Existence, uniqueness, regularity and simulations, Computers and Mathemtatics with Applications (2015), 1887—1905.  Con María Emilia Castillo. 

    Abstract: We perform a mathematical analysis of a model for drug dissolution–diffusion in non erodible nor swellable devices. We deduce a model and obtain a coupled nonlinear system which contains a parabolic equation for the dissolved drug and an ordinary differential equation for the solid drug, which is assumed to be distributed in the whole domain into microspheres which can differ in size. We analyze existence, uniqueness, and regularity properties of the system. Existence is proved using Schauder fixed point theorem. Lack of uniqueness is shown when the initial concentration of dissolved drug is higher than the saturation density in a region, and uniqueness is obtained in the non-saturated case. A square root function appears in the equation for the solid drug, and is responsible for the lack of uniqueness in the oversaturated case. The regularity results are sufficient for the optimal a priori error estimates of a finite element discretization of the system, which is presented and analyzed here. Simulations illustrating some features of the solutions and a good agreement with laboratory experiments are presented. Finally, we obtain error estimates for the finite element method used to compute the simulations.

    Keywords: Dissolution–diffusion, Drug release, Partial differential equations, Finite elements

    Artículo completo

  14. A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces. ESAIM Math. Model. Numer. Anal. 48 (2014), no. 6, 1557—1581.  Con Juan Pablo Agnelli y Eduardo M. Garau. 

    Abstract: In this article we develop a posteriori error estimates for general second order elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt's class $A_2$. The theory hinges on local approximation properties of either Clément or Scott-Zhang interpolation operators, without need of suitable modifications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good effectivity indices.

    Keywords: elliptic problems, point sources, a posteriori error estimates, finite elements, weighted Sobolevspaces

    AMS Subject Classifications: 35J15 / 65N12 / 65N15 / 65N30 / 65N50 / 65Y20

    Artículo completo

  15. Approximation classes for adaptive higher order finite element approximation, Mathematics of Computation 83 (2014), 2127—2160.  Con Fernando D. Gaspoz. 

    Abstract: We provide an almost characterization of the approximation classes appearing when using adaptive finite elements of Lagrange type of any fixed polynomial degree. The characterization is stated in terms of Besov regularity, and requires the approximation within spaces with integrability indices below one. This article generalizes to higher order finite elements the results presented for linear finite elements by Binev et. al. "Approximation Classes for Adaptive Methods", Serdica Math. J. 28 (2002), pp. 391-416.

    Keywords: Adaptive finite elements, Besov spaces, convergence rates, approximation classes

    AMS Subject Classifications: Primary 41A25, 65D05; Secondary 65N30, 65N50

    Artículo completo
    Errata

  16. AFEM for Geometric PDE: The Laplace-Beltrami Operator, in "Analysis and Numerics of Partial Differential Equations", edited by F. Brezzi, P. Colli Franzone, U. Gianazza, G. Gilardi. Springer INdAM Series, Vol. 4. 2012. 257—306.  Con Andrea Bonito, J. Manuel Cascón y Ricardo H. Nochetto. 

    Abstract: We present several applications governed by geometric PDE, and their parametric finite element discretization, which might yield singular behavior. The success of such discretization hinges on an adequate variational formulation of the Laplace-Beltrami operator, which we describe in detail for polynomial degree 1. We next present a complete a posteriori error analysis which accounts for the usual PDE error as well as the geometric error induced by interpolation of the surface. This leads to an adaptive finite element method (AFEM) and its convergence. We discuss a contraction property of AFEM and show its quasi-optimal cardinality.

    Artículo completo

  17. Adaptive finite element method for shape optimization. ESAIM: Control, Optimisation and Calculus of Variations 18 (2012), 1122—1149.  Con Ricardo H. Nochetto, M. Sebastián Pauletti y Marco Verani . 

    Abstract: We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and \bnew{thus} optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution---a new paradigm in adaptivity.

    Keywords: Shape optimization, adaptivity, mesh refinement/coarsening, smoothing

    AMS Subject Classifications: 49M25,65M60

    Artículo completo

  18. On uniform consistent estimators for convex regression}, Journal of Nonparametric Statistics 23 % issue 4 (2011), 897—908.  Con Néstor E. Aguilera y Liliana Forzani. 

    Abstract: A new nonparametric estimator of a convex regression function in any dimension is proposed and its uniform convergence properties are studied. We start by using any estimator of the regression function and convexify it by taking the convex envelope of a sample of the approximation obtained. We prove that the uniform rate of convergence of the estimator is maintained after the convexification is applied. The finite-sample properties of the new estimator are investigated by means of a simulation study and the application of the new method is demonstrated in examples.

    Keywords: approximation, convex regression, convexity, data-smoothing, nonparametric regression

    AMS Subject Classifications: 62G08, 62H12

    Artículo completo

  19. Quasi-optimal convergence rate of an AFEM for quasi-linear problems, Numerical Mathematics: Theory, Methods and Applications (2012), 131—156.  Con Eduardo M. Garau y Carlos Zuppa. 

    Abstract: We prove the quasi-optimal convergence of a standard adaptive finite ele- ment method (AFEM) for a class of nonlinear elliptic second-order equations of mono- tone type. The adaptive algorithm is based on residual-type a posteriori error estimators and Dörfler’s strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, analogous to the one used by Diening and Kreuzer (2008) and equivalent to the total error defined by Cascón et. al. (2008). This con- traction implies linear convergence of the discrete solutions to the exact solution in the usual H1 Sobolev norm. Secondly, we use this contraction to derive the optimal complexity of the AFEM. The results are based on ideas from Diening and Kreuzer and extend the theory from Cascón et. al. to a class of nonlinear problems which stem from strongly monotone and Lipschitz operators.

    Keywords: Adaptive Finite Element Methods , Optimality , Quasilinear Elliptic Equations

    AMS Subject Classifications: 35J62, 65N30, 65N12

    Artículo completo

  20. Convergence of an adaptive Kačanov FEM for quasi-linear problems, Applied Numerical Mathematics, 61 (2011), 512—529.  Con Eduardo M. Garau y Carlos Zuppa. 

    Abstract: We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Kačanov iteration and a mesh adaptation step is performed after each linear solve. The method is thus inexact because we do not solve the discrete nonlinear problems exactly, but rather perform one iteration of a fixed point method (Kačanov), using the approximation of the previous mesh as an initial guess. The convergence of the method is proved for any reasonable marking strategy and starting from any initial mesh. We conclude with some numerical experiments that illustrate the theory.

    Keywords: Nonlinear stationary conservation lawsAdaptive finite element methodsConvergence

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  21. Preconditioning a Class of Higher Order Operators by Operator Splitting, Numerische Mathematik (2011), 197—228.  Con Eberhard Bänsch y Ricardo H. Nochetto. 

    Abstract: We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H^s-regularity, s > 1, of the continuous problem nor quasi-uniformgrids.

    AMS Subject Classifications: 65F08, 65N22

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  22. AFEM for the Laplace-Beltrami Operator on Graphs: Design and Conditional Contraction Property, Mathematics of Computation, 80 (2011) 625—648.  Con Khamron Mekchay y Ricardo H. Nochetto. 

    Abstract: We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on $C^1$ graphs $\Gamma$ in $\R^d ~(d\ge2)$. We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in $H^1(\Gamma)$ and the surface error in $W^1_\infty(\Gamma)$ due to approximation of $\Gamma$. We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of $\Gamma$ in $W^1_\infty$. We conclude with one numerical experiment that illustrates the theory.

    Keywords: Laplace-Beltrami operator, graphs, adaptive finite element method, a posteriori error estimate, energy and geometric errors, bisection, contraction.

    Artículo completo

  23. Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA Journal of Numerical Analysis (2011) 31(3): 914—946.  Con Eduardo M. Garau. 

    Abstract: In this article we prove convergence of adaptive finite element methods for Steklov eigenvalue problems under very general assumptions for simple as well as multiple eigenvalues starting from any initial triangulation. We also prove the optimality of the approximations assuming Dörfler's Strategy for marking, when we consider simple eigenvalues.

    Keywords: Steklov Eigenvalue problems; adaptive finite element methods; convergence; optimality.

    Artículo completo

  24. On convex functions and the finite element method, SIAM J. Numer. Anal., 47 (2009), 3139—3157 .  Con Néstor E. Aguilera. 

    Abstract: Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.

    Keywords: Finite element method, optimization problems, convex functions, adaptive meshes.

    AMS Subject Classifications: 65K10,65N30.

    Artículo completo

  25. Convergence of adaptive finite element methods for eigenvalue problems, Mathematical Models and Methods in Applied Sciences, Vol. 19, No. 5 (2009), 721—747.  Con Eduardo M. Garau y Carlos Zuppa. 

    Abstract: In this article we prove convergence of adaptive finite element methods for second order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation.

    Keywords: eigenvalue problems, adaptive finite elements, convergence, optimality, regularity

    Artículo completo

  26. Approximating optimization problems over convex functions, Numerische Mathematik 111 (2008), 1—34.  Con Néstor E. Aguilera. 

    Abstract: Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using positive semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.

    Keywords:

    AMS Subject Classifications: 52B55 - 65D15 - 90C90

    Artículo completo

  27. Convergence rates for adaptive finite element methods, IMA Journal of Numerical Analysis 29 (2009), 917—936.  Con Fernando D. Gaspoz. 

    Abstract: In this article we prove that it is possible to construct, using newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite element methods (AFEM) using Lagrange finite elements of any polynomial degree are obtained.

    Keywords: convergence rates, adaptive finite elements, optimality, regularity

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  28. A variational shape optimization approach for image segmentation with a Mumford-Shah functional, SIAM Journal on Scientific Computing 30 (2008), 3028—3049.   Con Gunay Dögan y Ricardo H. Nochetto. 

    Abstract: We introduce a novel computational method for a Mumford-Shah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields non-linear PDE flows. We propose time discretizations that linearize the problem, and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, space-time adaptivity and a coarse-to-fine approach to process large images efficiently. We present several simulations that illustrate the performance of the method, and investigate the model sensitivity to various parameters.

    Keywords: image segmentation, Mumford-Shah, shape optimization, finite element method

    AMS Subject Classifications: 49M15,49M25,65D15,65K10,68T45,90C99

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  29. A basic convergence result for conforming adaptive finite elements, Mathematical Models and Methods in the Applied Sciences 18 (2008), 707—737.  Con Kunibert G. Siebert y Andreas Veeser 

    Abstract: We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of `saddle point' type. For the adaptive algorithm we suppose the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf-sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dörfler's strategy, but also by the maximum strategy and the equidistribution strategy.

    Keywords: Adaptivity, conforming finite elements, convergence

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  30. Discrete Gradient Flows for Shape Optimization and Applications, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3898—3914.  Con Gunay Dögan, Ricardo H. Nochetto y Marco Verani. 

    Abstract: We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.

    Keywords: Shape optimization, scalar product, gradient flow, semi-implicit discretization, finite elements, surface diffusion, image segmentation.

    AMS Subject Classifications: 49Q10, 65M60, 65K10

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  31. A Finite Element Method for Surface Diffusion: the Parametric Case, Journal of Computational Physics 203 (2005), 321—343.  Con Eberhard Bänsch y Ricardo H. Nochetto. 

    Abstract: Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for parametric surfaces with or without boundaries. The method is semi-implicit, requires no explicit parametrization, and yields a linear system of elliptic PDE to solve at each time step. We next develop a finite element method, propose a Schur complement approach to solve the resulting linear systems, and show several significant simulations, some with pinch-off in finite time. We introduce a mesh regularization algorithm, which helps prevent mesh distortion, and discuss the use of time and space adaptivity to increase accuracy while reducing complexity.

    Keywords: Surface diffusion, fourth-order parabolic problem, finite elements, Schur complement, smoothing effect, pinch-off.

    AMS Subject Classifications: 35K55, 65M12, 65M15

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  32. Surface Diffusion of Graphs: Variational Formulation, Error Analysis and Simulation, SIAM Journal on Numerical Analysis 42 (2004), 773—799.  Con Eberhard Bänsch y Ricardo H. Nochetto. 

    Abstract: Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for graphs and derive a priori error estimates for a time-continuous finite element discretization. We also introduce a semi-implicit time discretization and a Schur complement approach to solve the resulting fully discrete, linear systems. After computational verification of the orders of convergence for polynomial degrees 1 and 2, we show several simulations in 1d and 2d with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time, such as infinite slopes and cracks.

    Keywords: Surface diffusion, fourth-order parabolic problem, finite elements, a priori error estimates, Schur complement, smoothing effect.

    AMS Subject Classifications: 35K55, 65M12, 65M15, 65M60, 65Z05.

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  33. Convergence of Adaptive Finite Element Methods, SIAM Review 44 (2002), 631—658  Con Ricardo H. Nochetto y Kunibert G. Siebert. 

    Abstract: Adaptive finite element methods (FEM) have been widely used in applications for over 20 years now. In practice, they converge starting from coarse grids, although no mathematical theory has been able to prove this assertion. Ensuring an error reduction rate based on a posteriori error estimators, together with a reduction rate of data oscillation (information missed by the underlying averaging process), we construct a simple and efficient adaptive FEM for elliptic partial differential equations. We prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in two and three dimensions yield quasi-optimal meshes along with a competitive performance. Extensions to higher order elements and applications to saddle point problems are discussed as well.

    Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, Stokes, Uzawa

    AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20

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  34. An Adaptive Uzawa FEM for the Stokes problem: Convergence without the Inf-Sup condition, SIAM Journal on Numerical Analysis~40 (2002), 1207—1229.  Con Eberhard Bänsch y Ricardo H. Nochetto. 

    Abstract: We introduce and study an adaptive finite element method for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity whereas for pressure the elements can be either discontinuous of degree k-1 or continuous of degree k-1 and k. The popular Taylor-Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver, and provide consistent computational evidence that the resulting meshes are quasi-optimal.

    Keywords: A posteriori error estimators, adaptive mesh refinement, convergence, data oscillation, performance, quasi-optimal meshes

    AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20

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  35. Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance, Mathematics of Computation 72 (2003), 1067—1097.  Con Ricardo H. Nochetto y Kunibert G. Siebert. 

    Abstract: A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh preadaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.

    Keywords: A posteriori error estimators, local problems, stars, data oscillation, adaptivity, convergence, performance

    AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20

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  36. Data Oscillation and Convergence of Adaptive FEM, SIAM Journal on Numerical Analysis 38 (2000), 466—488. Con Ricardo H. Nochetto y Kunibert G. Siebert. 

    Abstract: Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic PDE with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in 2d and 3d yield quasi-optimal meshes along with a competitive performance.

    Keywords: A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, performance, quasi-optimal meshes

    AMS Subject Classifications: 65N12, 65N15, 65N30, 65N50, 65Y20

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  37. Parameter Identification for Nonlinear Abstract Cauchy Problems using Quasilinearization, Journal of Optimization Theory and Applications 113 (2002), no. 2, 227—250.  Con Rubén D. Spies  

    Abstract: An approach to quasilinearization for parameter identification in nonlinear abstract Cauchy problems in which the parameter appears in the nonlinear term, is presented. This new approach has two main advantages over the classical one: it is much more intuitive and the derivation of the algorithm is done without need of the sensitivity equations on which classical quasilinearization is based upon. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. A comparison with the standard approach is presented and an application example is shown in which the non-physical parameters in a mathematical model for shape memory alloys are estimated.

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  38. Parameter Differentiability of the Solution of a Nonlinear Abstract Cauchy Problem, Journal of Mathematical Analysis and Applications 252 (2000), 18—31.  Con John A. Burns y Rubén D. Spies. 

    Abstract: The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions (martensitic transformations) in one-dimensional Shape Memory Alloys (SMA) with non-convex Landau-Ginzburg free energy potentials. This system is formally written as a nonlinear abstract Cauchy problem in an appropriate Hilbert Space. A quasilinearization-based algorithm for parameter identification in this kind of Cauchy problems is proposed. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. Numerical examples are presented in which the algorithm is applied to recover the non-physical parameters describing the free energy potential in SMA, both from exact and noisy data.

    Keywords: Parameter Identification, Shape Memory Alloys, Free Energy. Partial Differential Equations, Abstract Cauchy Problems, Quasilinearization.

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  39. A Quasilinearization Approach for Parameter Identification in a Nonlinear Model of Shape Memory Alloys, Inverse Problems 14 (1998), 1551—1563. Con Rubén D. Spies. 

    Abstract: The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions (martensitic transformations) in one-dimensional Shape Memory Alloys (SMA) with non-convex Landau-Ginzburg free energy potentials. This system is formally written as a nonlinear abstract Cauchy problem in an appropriate Hilbert Space. A quasilinearization-based algorithm for parameter identification in this kind of Cauchy problems is proposed. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. Numerical examples are presented in which the algorithm is applied to recover the non-physical parameters describing the free energy potential in SMA, both from exact and noisy data.

    Keywords: Parameter Identification, Shape Memory Alloys, Free Energy. Partial Differential Equations, Abstract Cauchy Problems, Quasilinearization.

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  40. Modal Approximations for the Dynamics of Shape Memory Alloys Under External Thermomechanical Actions, Latin American Applied Research 29 (1999), 151—159. Con Rubén D. Spies. 

    Abstract: In this article an algorithm for numerically solving the nonlinear system of partial differential equations that model the dynamics of martensitic phase transitions in one-dimensional Shape Memory Alloys is presented. The algorithm is based upon a state-space formulation of the equations. The approximations are defined in terms of the eigenvalues and eigenvectors of the operator associated to the linear part of the resulting semilinear Cauchy problem. For the alloy Au$_{23}$Cu$_{30}$Zn$_{47}$ numerical results are shown under the effect of different external distributed actions and for several initial conditions.

    Keywords: Shape Memory Alloys, hysteresis, conservation laws, initial-boundary value problem, spectral approximations, modal approximations.

    AMS Subject Classifications: 35A35, 35A40, 35M05, 73U05, 73C35.

  41. Convergent Spectral Approximations for the Thermomechanical Processes in Shape Memory Alloys, Nonlinear Analysis: Theory, Methods, and Applications 39(2000), 11—32.  Con Terry L. Herdman y Rubén D. Spies. 

    Abstract: In this article discrete spectral approximations to the nonlinear evolutionary partial differential equations that model the dynamics of thermomechanical solid-solid phase transitions in one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are constructed. By using the theories of analytic semigroups and interpolation spaces and a generalization of Gronwall's lemma for singular kernels, the convergence of the approximations is proved. For the alloy Au$_{23}$Cu$_{30}$Zn$_{47}$ numerical results are shown under different external distributed actions and initial conditions.

    Keywords: Shape Memory Alloys, non-convex potential, hysteresis, conservation laws, initial-boundary value problem, spectral approximations.

    AMS Subject Classifications: 35A35, 35A40, 35M05, 73U05, 73C35.

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  42. Identifiability of the Landau-Ginzburg Potential in a Mathematical Model of Shape Memory Alloys, Journal of Mathematical Analysis and Applications 212 (1997), 292—315.  Con Rubén D. Spies. 

    Abstract: The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions in one-dimensional shape memory solids with non-convex Landau-Ginzburg free energy potentials. In this article the theories of analytic semigroups and real interpolation spaces for maximal accretive operators are used to show that the solutions of the model depend continuously on the admissible parameters. Also, we show that the non-physical parameters that define the free energy are identifiable from the model.

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  43. Parameter Continuity of the Solutions of a Mathematical Model of Thermoviscoelasticity, Revista de la Unión Matemática Argentina 40 (1996), 111—126.  con Rubén D. Spies. 

    Abstract: In this paper the continuity of the solutions of a mathematical model of thermoviscoelasticity with respect to the model parameters is proved. This was an open problem conjectured in [27] and [28]. The nonlinear partial differential equations under consideration arise from the conservation laws of linear momentum and energy and describe structural phase transitions in solids with non-convex Landau-Ginzburg free energy potentials. The theories of analytic semigroups and real interpolation spaces for maximal accretive operators are used to show that the solutions of the model depend continuously on the admissible parameters, in particular, on those defining the free energy. More precisely, it is shown that if $\{q_n\}_{n=1}^{\infty}$ is a sequence of admissible parameters converging to $q$, then the corresponding solutions $z(t;q_n)$ converge to $z(t;q)$ in the norm of the graph of a fractional power of the operator associated to the linear part of the system.
    [27] SPIES, R. D.; A State-Space Approach to a One-Dimensional Mathematical Model for the Dynamics of Phase Transitions in Pseudoelastic Materials, Journal of Mathematical Analysis and Applications 190 (1995), 58-100.
    [28] SPIES, R. D.; Results on a Mathematical Model of Thermomechanical Phase Transitions in Shape Memory Materials, Journal of Smart Materials and Structures 3 (1994), 459-469.

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