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Pedro Morin

Publications and source records attributed to Pedro Morin.

At least 19 recordsLinked to original sources
arXiv2026

Error estimates for the patch bubble method for convection-dominated channel flow problem

We present error estimates for the BMZ (Bubble Mesh Zoom) residual-free bubble method applied to a convection-diffusion equation in the convection-dominated regime. The method incorporates both element bubbles and residual-free bubbles supported on patches of two adjacent elements. We focus on the case of a parallel flow in a square domain and derive error estimates in an energy norm that remain valid as diffusion becomes small. The theoretical findings are corroborated by numerical experiments, which also exhibit a competitive performance of the method.

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arXiv2026

Approximation classes for the anisotropic space-time finite element method. An almost characterization

We study the approximation of $L_p$-functions, $p\in (0,\infty]$, on cylindrical space-time domains $\Omega_T:=[0,T]\times \Omega$, $0<T<\infty$, $\Omega\subset \R^d$ Lipschitz, $d\in \mathbb{N}$, with respect to continuous anisotropic space-time finite elements on prismatic meshes. In particular, we propose a suitable refinement technique which creates (locally refined) prismatic meshes with sufficient smoothness and the desired anisotropy, and prove complexity estimates. Furthermore, we define a (quasi-)interpolation operator on this type of meshes and use it to characterize the corresponding approximation classes by showing direct and inverse estimates in terms of anisotropic Besov norms.

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arXiv2025

Anisotropic approximation on space-time domains

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spacial moduli of smoothness and their properties. In particular, we prove Jackson- and Whitney-type inequalities on Lipschitz cylinders, i.e., space-time domains $I\times D$ with a finite interval $I$ and a bounded Lipschitz domain $D\subset \R^d$, $d\in \N$. As an application, we prove a direct estimate result for adaptive space-time finite element approximation in the discontinuous setting.

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arXiv2025

Patch bubbles for advection-dominated steady and unsteady problems

A novel variant of the \emph{residual-free bubble} method (RFB) for advection dominated problems is presented. Since the usual RFB still suffers from oscillations and strong under/overshoots, the bubble space is enriched by \emph{patch bubbles}, giving more freedom to the bubble space. We use a recursive and efficient approach to accurately compute the bubbles. Numerical experiments clearly demonstrate the superiority of our method compared to the standard RFB. The method is similar to the \emph{enhanced residual-free bubble} method (eRFB) proposed by Cangiani and S\"uli in 2005, but differs in the definition of the additional bubbles.

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arXiv2025

A convergent algorithm for mean curvature flow of surfaces with Dirichlet boundary conditions

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling precise control of discretization errors and yielding optimal error estimates for discrete spaces with piecewise polynomials of degree $p \geq 2$. Building on techniques recently developed by Kov\'acs, Li, Lubich, and collaborators for closed surfaces, we extend these ideas to surfaces with boundaries by formulating appropriate boundary conditions for both the mean curvature and the normal vector. These boundary treatments are essential for proving convergence. The core of our analysis involves a classical error splitting strategy using auxiliary discrete functions that approximate the surface geometry, the mean curvature, and the normal vector. We estimate two types of errors for each variable to rigorously assess both stability and consistency. To effectively handle boundary conditions for the normal vector, we introduce a nonlinear Ritz projection into the analysis. As a result, we derive optimal $H^1$ error estimates for the surface position, velocity, mean curvature, and normal vector. Our theoretical findings are corroborated by numerical experiments.

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arXiv2023

Error analysis for local coarsening in univariate spline spaces

In this article we analyze the error produced by the removal of an arbitrary knot from a spline function. 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 control points 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.

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arXiv2021

Approximation classes for adaptive time-stepping finite element methods

We study approximation classes for adaptive time-stepping finite element methods for time-dependent Partial Differential Equations (PDE). We measure the approximation error in $L_2([0,T)\timesΩ)$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a byproduct we define Besov spaces for vector-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.

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arXiv2019

Convective transport in nanofluids: regularity of solutions and error estimates for finite element approximations

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.

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arXiv2018

A new perspective on hierarchical spline spaces for adaptivity

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. Such framework makes it simple to create hierarchical \emph{basis} with \emph{control on the overlapping}. Linear independence is always desired for the well posedness of the linear systems, and to avoid redundancy. The control on the overlapping of basis functions from different levels is necessary to close theoretical arguments in the proofs of optimality of adaptive methods. In order to guarantee linear independence, and to control the overlapping of the basis functions, some basis functions additional to those initially marked must be refined. However, with our framework and refinement procedures, the complexity of the resulting bases is under control, i.e., the resulting bases have cardinality bounded by the number of initially marked functions.

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arXiv2017

The shape derivative of the Gauss curvature

We introduce new results about the shape derivatives of scalar- and vector-valued functions, extending the results from (Dogan-Nochetto 2012) to more general surface energies. They 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 introduce a new scheme of Newton-type to approximate a minimizer of a shape functional. It is a mathematically sound generalization of the method presented in (Chicco-Ruiz et. al. 2016). 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 Newton-type method mentioned above.

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arXiv2015

High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ 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 $W^1_\infty$ 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 $W^1_\infty$ and PDE error in $H^1$.

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arXiv2015

A posteriori error estimates with point sources in fractional Sobolev spaces

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 decay rates.

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arXiv2010

Quasi-optimal convergence rate of an AFEM for quasi-linear problems

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and D\"orfler's strategy is assumed for marking. We first prove a contraction property for a suitable definition of total error, which is equivalent to the total error as defined by Casc\'on et al. (in SIAM J. Numer. Anal. 46 (2008), 2524--2550), and implies linear convergence of the algorithm. Secondly, we use this contraction to derive the optimal cardinality of the AFEM.

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arXiv2010

Convergence of an adaptive Ka\v{c}anov FEM for quasi-linear problems

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

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arXiv2010

On convex regression estimators

A new nonparametric estimator of a convex regression function in any dimension is proposed and its convergence properties are studied. We start by using any estimator of the regression function and we \emph{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.

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arXiv2008

Approximating optimization problems over convex functions

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 optimizing functionals arising from 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 semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.

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arXiv2008

On convex functions and the finite element method

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.

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