Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems
type: Item
instance of: scholarly article
title: Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems
zbMATH Open document ID: 1268.65138
author: Thomas Richter
author: Andreas Springer
author: Boris Vexler
published in: Numerische Mathematik
publication date: 2013-05-13T00:00:00Z
review text: The authors propose an efficient numerical realization of discontinuous Galerkin methods (dG) for temporal discretization of parabolic partial differential equations of the form \(\partial_t u(t) + A(u) = f(t)\),\; \(t \in (0,T)\), with initial condition \(u(0) = u_0\). Two sets of assumptions are imposed on the spatial differential operator that guarantee the existence of a unique solution and the applicability of the proposed simplified solution scheme. A suitable approximation to Newton's method and decoupling the Newton update equation, which consists of a coupled system of \(r+1\) elliptic problems will avoid complex coefficients which arise inevitably in the equations obtained by a direct decoupling. The accuracy of these approximations and their impact on the outer Newton iteration are investigated. The efficiency of the presented schemes is studied by numerical tests.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 6162441
zbMATH Keywords: discontinuous Galerkin method
zbMATH Keywords: temporal discretization
zbMATH Keywords: parabolic problem
zbMATH Keywords: numerical examples
zbMATH Keywords: Newton's method
describes a project that uses: RADAU
describes a project that uses: GASCOIGNE
describes a project that uses: RoDoBo
MaRDI profile type: Publication
full work available at URL: s00211-012-0511-7
OpenAlex ID: W2070144563
cites work: Efficient numerical solution of parabolic optimization problems by finite element methods
cites work: Q4840765
cites work: Q5439726
cites work: The finite element methods for elliptic problems.
cites work: Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem
cites work: Adaptive Finite Element Methods for Parabolic Problems II: Optimal Error Estimates in $L_\infty L_2 $ and $L_\infty L_\infty $
cites work: Time discretization of parabolic problems by the discontinuous Galerkin method
cites work: Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems
cites work: Stiff differential equations solved by Radau methods
cites work: Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation
cites work: Adaptive multilevel solution of nonlinear parabolic PDE systems. Theory, algorithm, and applications
cites work: A Priori Error Analysis of the Petrov–Galerkin Crank–Nicolson Scheme for Parabolic Optimal Control Problems
cites work: Adaptive Space‐Time Finite Element Methods for Parabolic Optimization Problems
cites work: A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints
cites work: A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part II: Problems with Control Constraints
cites work: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems
cites work: Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations
cites work: Adaptivity with Dynamic Meshes for Space-Time Finite Element Discretizations of Parabolic Equations
cites work: Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method
cites work: A posteriori error estimation for \(hp\)-version time-stepping methods for parabolic partial differential equations
cites work: \(hp\)-discontinuous Galerkin time stepping for parabolic problems
cites work: Galerkin Finite Element Methods for Parabolic Problems
cites work: Order stars and stability theorems
cites work: Realistic Eigenvalue Bounds for the Galerkin Mass Matrix
cites work: Q3760934
cites work: Local Convergence of Inexact Newton Methods
DOI: 10.1007/S00211-012-0511-7
description: scientific article; zbMATH DE number 6162441