Computational Methods in Applied Mathematics

Published by De Gruyter

Volume 20 Issue 2

Issue of Computational Methods in Applied Mathematics

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Contents
Journal Overview

Publicly Available April 1, 2020

Frontmatter

Page range: i-iv

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Publicly Available April 9, 2019

A Fully-Mixed Finite Element Method for the n-Dimensional Boussinesq Problem with Temperature-Dependent Parameters

Javier A. Almonacid, Gabriel N. Gatica

Page range: 187-213

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Abstract

In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n -dimensional fluids, n∈{2,3}{n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported.

August 14, 2019

Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem

Fleurianne Bertrand, Daniele Boffi, Rolf Stenberg

Page range: 215-225

Abstract

This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard H01{H_{0}^{1}}-conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches.

June 19, 2019

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

Michele Botti, Daniele A. Di Pietro, Pierre Sochala

Page range: 227-249

Abstract

In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

May 7, 2019

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Christoph Erath, Robert Schorr

Page range: 251-272

Abstract

Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

August 28, 2019

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

Thirupathi Gudi, Papri Majumder

Page range: 273-292

Abstract

We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order 𝒪⁢(h+Δ⁢t){\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity ut∈L2⁢(0,T;L2⁢(Ω)){u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

Publicly Available June 13, 2019

Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation

Johannes Kraus, Svetoslav Nakov, Sergey I. Repin

Page range: 293-319

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Abstract

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

March 8, 2019

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

Pallavi Mahale, Sharad Kumar Dixit

Page range: 321-341

Abstract

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

June 19, 2019

Convergence and Preconditioning of Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

Rayan Nasser, Miloud Sadkane

Page range: 343-359

Abstract

This paper focuses on the inner iteration that arises in inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil. First, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy. Then, as inner iteration, block-GMRES is used with preconditioners generalizing the one by Robbé, Sadkane and Spence [Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 31 2009, 1, 92–113]. It is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately a small constant. The efficiency of the preconditioners is illustrated by numerical examples.

February 15, 2019

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Tamal Pramanick, Rajen Kumar Sinha

Page range: 361-378

Abstract

The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪(HsLog^(Hh)s2){\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪(H2⁢sLog^(Hh)s){\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L∞⁢(H1){L^{\infty}(H^{1})}-norm and L∞⁢(L2){L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

June 13, 2019

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

Tao Wang, Chaochao Yang, Xiaoping Xie

Page range: 379-393

Abstract

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the L2{L^{2}} norm are derived. Numerical results are provided to verify the theoretical results.

Publicly Available March 28, 2020

Retraction of: Stabilizability of Infinite-Dimensional Systems by Finite-Dimensional Controls

The Editors

Page range: 395-395

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About this journal

Objective
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.

CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.

Topics

Numerical and computational mathematics
Partial differential equations
Applied mathematics

Article formats
Original research articles

Proposals for special issues of CMAM are considered.
Note that for special issue proposals not only an exciting topic within the scientific scope of the journal is required, but also the Guest Editors need to have an outstanding worldwide reputation in their field.

CMAM announces the preparation of a special issue on "Numerical Methods for PDEs" dedicated to the memory of Professor Raytcho Lazarov, who passed away on 30 November, 2024. Guest Editors: Svetozar Margenov and Ulrich Langer. All submissions will be reviewed following the general editorial policy of the journal. The submission deadline is 31 August, 2025. Please submit your manuscript online at https://mc.manuscriptcentral.com/cmam