Finite Element Method (Fem)
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A FETI Method with a Mesh Independent Condition Number for the Iteration Matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera
A FETI method with a mesh independent condition number for the iteration matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera To cite this version: Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera. A FETI method with a mesh in- dependent condition number for the iteration matrix. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (issues-13-16), pp.1410-1429. 10.1016/j.cma.2007.11.019. hal- 00170169 HAL Id: hal-00170169 https://hal.archives-ouvertes.fr/hal-00170169 Submitted on 6 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A FETI method with a mesh independent condition number for the iteration matrix C. Bernardi1, T . Chac´on Rebollo2, E. Chac´on Vera2 August 21, 2007 Abstract. We introduce a framework for FETI methods using ideas from the 1 decomposition via Lagrange multipliers of H0 (Ω) derived by Raviart-Thomas [22] and complemented with the detailed work on polygonal domains devel- oped by Grisvard [17]. We compute the action of the Lagrange multipliers 1/2 using the natural H00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration ma- trix is independent of the mesh size and there is no need for preconditioning. -
Hp-Finite Element Methods for Hyperbolic Problems
¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Z¨urich Swiss Federal Institute of Technology Zurich hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Research Report No. 99-14 July 1999 Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland †Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom ∗E. S¨uliand P. Houston acknowledge the financial support of the EPSRC (Grant GR/K76221) hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland Research Report No. 99-14 July 1999 Abstract This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial dif- ferential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equa- tions and second-order problems of mixed elliptic-hyperbolic-parabolic type. An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size h; the error bound is either 1 degree or 1/2 de- gree below optimal in terms of the polynomial degree p, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively. In the case of a first-order hyperbolic equation the error bound is hp-optimal in the DG-norm. -
An Adaptive Hp-Refinement Strategy with Computable Guaranteed Bound
An adaptive hp-refinement strategy with computable guaranteed bound on the error reduction factor∗ Patrik Danielyz Alexandre Ernzy Iain Smearsx Martin Vohral´ıkyz April 24, 2018 Abstract We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div; Ω)-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via D¨orfler’s bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed hp-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one. Key words: a posteriori error estimate, adaptivity, hp-refinement, finite element method, error reduction, equilibrated flux, residual lifting 1 Introduction Adaptive discretization methods constitute an important tool in computational science and engineering. -
Newton-Krylov-BDDC Solvers for Nonlinear Cardiac Mechanics
Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics Item Type Article Authors Pavarino, L.F.; Scacchi, S.; Zampini, Stefano Citation Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics 2015 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI 10.1016/j.cma.2015.07.009 Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 18 July 2015. DOI:10.1016/ j.cma.2015.07.009 Download date 07/10/2021 05:34:35 Link to Item http://hdl.handle.net/10754/561071 Accepted Manuscript Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics L.F. Pavarino, S. Scacchi, S. Zampini PII: S0045-7825(15)00221-2 DOI: http://dx.doi.org/10.1016/j.cma.2015.07.009 Reference: CMA 10661 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 13 December 2014 Revised date: 3 June 2015 Accepted date: 8 July 2015 Please cite this article as: L.F. Pavarino, S. Scacchi, S. Zampini, Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication. -
Finite Difference and Discontinuous Galerkin Methods for Wave Equations
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. -
Combining Perturbation Theory and Transformation Electromagnetics for Finite Element Solution of Helmholtz-Type Scattering Probl
Journal of Computational Physics 274 (2014) 883–897 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Combining perturbation theory and transformation electromagnetics for finite element solution of Helmholtz-type scattering problems ∗ Mustafa Kuzuoglu a, Ozlem Ozgun b, a Dept. of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkey b Dept. of Electrical and Electronics Engineering, TED University, 06420 Ankara, Turkey a r t i c l e i n f o a b s t r a c t Article history: A numerical method is proposed for efficient solution of scattering from objects with Received 31 August 2013 weakly perturbed surfaces by combining the perturbation theory, transformation electro- Received in revised form 13 June 2014 magnetics and the finite element method. A transformation medium layer is designed over Accepted 15 June 2014 the smooth surface, and the material parameters of the medium are determined by means Available online 5 July 2014 of a coordinate transformation that maps the smooth surface to the perturbed surface. The Keywords: perturbed fields within the domain are computed by employing the material parameters Perturbation theory and the fields of the smooth surface as source terms in the Helmholtz equation. The main Transformation electromagnetics advantage of the proposed approach is that if repeated solutions are needed (such as in Transformation medium Monte Carlo technique or in optimization problems requiring multiple solutions for a set of Metamaterials perturbed surfaces), computational resources considerably decrease because a single mesh Coordinate transformation is used and the global matrix is formed only once. -
A Discontinuous Galerkin Method for Nonlinear Parabolic Equations and Gradient flow Problems with Interaction Potentials
A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials Zheng Sun1, Jos´eA. Carrillo2 and Chi-Wang Shu3 Abstract We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker-Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics. Keywords: discontinuous Galerkin method, positivity-preserving, entropy-entropy dissipation relationship, nonlinear parabolic equation, gradient flow 1Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: zheng [email protected] 2Department of Mathematics, Imperial College London, London SW7 2AZ, UK. E-mail: car- [email protected]. Research supported by the Royal Society via a Wolfson Research Merit Award and by EPSRC grant number EP/P031587/1. 3Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected]. -
Robust Finite Element Solution
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 212 (2008) 374–389 www.elsevier.com/locate/cam An elliptic singularly perturbed problem with two parameters II: Robust finite element solutionଁ Lj. Teofanova,∗, H.-G. Roosb aDepartment for Fundamental Disciplines, Faculty of Technical Sciences, University of Novi Sad, Trg D.Obradovi´ca 6, 21000 Novi Sad, Serbia bInstitute of Numerical Mathematics, Technical University of Dresden, Dresden D-01062, Germany Received 8 November 2005; received in revised form 8 December 2006 Abstract In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis. © 2007 Elsevier B.V. All rights reserved. MSC: 35J25; 65N30; 65N15; 76R99 Keywords: Singular perturbation; Convection–diffusion problems; Finite element method; Two small parameters; Shishkin mesh 1. Introduction In this paper, we consider a finite element method for the following singularly perturbed elliptic two-parameter problem posed on the unit square = (0, 1) × (0, 1) Lu := −1u + 2b(x)ux + c(x)u = f(x,y) in , u = 0onj, (1) b(x) > 0, c(x) > 0 for x ∈[0, 1], (2) where 0 < 1, 2>1 and and are constants. We assume that b, c and f are sufficiently smooth functions and that f satisfies the compatibility conditions f(0, 0) = f(0, 1) = f(1, 0) = f(1, 1) = 0. -
Robust Adaptive Hp Discontinuous Galerkin Finite Element Methods For
Robust adaptive hp discontinuous Galerkin finite element methods for the Helmholtz equation∗ Scott Congrevey Joscha Gedickez Ilaria Perugiax Abstract This paper presents an hp a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree p and the wave number k. For the discretization, we consider a discontinuous Galerkin formulation that is unconditionally well posed. The a posteriori error analysis is based on the technique of equilibrated fluxes applied to a shifted Poisson problem, with the error due to the nonconformity of the discretization controlled by a potential reconstruction. We prove that the error estimator is both reliable and efficient, under the condition that the initial mesh size and polynomial degree is chosen such that the discontinuous Galerkin formulation converges, i.e., it is out of the regime of pollution. We confirm the efficiency of an hp-adaptive refinement strategy based on the presented robust a posteriori error estimator via several numerical examples. Keywords a posteriori error analysis, hp discontinuous Galerkin finite element method, equilibrated fluxes, potential reconstruction, Helmholtz problem AMS subject classification 65N15, 65N30, 65N50 1 Introduction In this paper, we consider the following Helmholtz problem with impedance boundary condition: Find a (complex) solution u 2 H2(Ω) such that −∆u − k2u = f in Ω; (1.1) ru · n − iku = g on @Ω; where Ω ⊂ R2 is a bounded, Lipschitz domain, n denotes the outer unit normal on the boundary @Ω, f 2 L2(Ω), g 2 L2(@Ω), and k > 0 is the (constant) wavenumber. The problem (1.1) was shown to be well-posed in [18]. -
Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods
Journal of Scientific Computing manuscript No. (will be inserted by the editor) Optimal strong-stability-preserving Runge–Kutta time discretizations for discontinuous Galerkin methods Ethan J. Kubatko · Benjamin A. Yeager · David I. Ketcheson Received: date / Accepted: date Abstract Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge– Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy (CFL) stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with re- spect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through ap- plication to numerical test cases. Keywords discontinuous Galerkin · Runge–Kutta · strong-stability-preserving E. Kubatko Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 2070 Neil Avenue, Columbus, OH 43210, United States Tel.: 614-292-7176 E-mail: [email protected] B. -
A Bddc Algorithm for the Stokes Problem with Weak 2 Galerkin Discretizations
1 A BDDC ALGORITHM FOR THE STOKES PROBLEM WITH WEAK 2 GALERKIN DISCRETIZATIONS 3 XUEMIN TU∗ AND BIN WANGy 4 Abstract. The BDDC (balancing domain decomposition by constraints) methods have been 5 applied successfully to solve the large sparse linear algebraic systems arising from conforming finite 6 element discretizations of second order elliptic and Stokes problems. In this paper, the Stokes 7 equations are discretized using the weak Galerkin method, a newly developed nonconforming finite 8 element method. A BDDC algorithm is designed to solve the linear system such obtained. Edge/face 9 velocity interface average and mean subdomain pressure are selected for the coarse problem. The 10 condition number bounds of the BDDC preconditioned operator are analyzed, and the same rate 11 of convergence is obtained as for conforming finite element methods. Numerical experiments are 12 conducted to verify the theoretical results. 13 Key words. Discontinuous Galerkin, HDG, weak Galerkin, domain decomposition, BDDC, 14 Stokes, Saddle point problems, benign subspace 15 AMS subject classifications. 65F10, 65N30, 65N55 16 1. Introduction. Numerical solution of saddle point problems using non over- 17 lapping domain decomposition methods have long been an active area of research; see, 18 e.g., [28, 15, 11, 10, 18, 29, 30, 16, 33, 17, 34, 27]. The Balancing Domain Decomposi- 19 tion by Constraints (BDDC) algorithm is an advanced variant of the non-overlapping 20 domain decomposition technique. It was first introduced by Dohrmann [5], and the 21 theoretical analysis was later given by Mandel and Dohrmann [20]. In this theoretical 22 development, optimal condition number bound was obtained for the BBDC opera- 23 tors proposed for symmetric positive definite systems. -
Weighted Scheme and the Galerkin Method
mathematics Article On the Numerical Simulation of HPDEs Using q-Weighted Scheme and the Galerkin Method Haifa Bin Jebreen *,† and Fairouz Tchier † Department of Mathematics, College of Science, King Saud University, Riyadh 11 551, Saudi Arabia; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the q-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis. Keywords: interpolating scaling functions; hyperbolic equation; Galerkin method 1. Introduction Partial differential equations (PDEs) are ubiquitous in mathematically scientific fields and play an important role in engineering and physics. They arise from many purely math- ematical considerations, such as the calculus of variations and differential geometry. One of Citation: Bin Jebreen, H.; Tchier, F. On the momentous subclasses of PDEs is the hyperbolic partial differential equations (HPDEs). the Numerical Simulation of HPDEs HPDEs are used to model many phenomena such as biology, industry, atomic physics, and Using q-Weighted Scheme and the aerospace [1–3]. Telegraph and wave equations are the most famous types of HPDEs that Galerkin Method.