Galerkin-Type Approximations which are Discontinuous in Time for Parabolic Equations in a Variable Domain

Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit Runge-KuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction U of the discrete solution U , which restores continuity and leads to the differential equation U +ΠF(U ) = F for a suitable interpolation operator Π. The error analysis hinges on careful energy arguments and the monotonicity of the operator F, in particular its angle bounded structure. We discuss applications to linear PDE such as the convection-diffusion equation and the wave equation, and nonlinear PDE corresponding to subgradient operators such as the p-Laplacian and minimal surfaces, as well as Lipschitz and noncoercive operators.

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“…The discontinuous Galerkin method dG(q) of order q ≥ 1 is defined as follows [7,8,9,10,11,14,15,16,17,21,14,30]:…”

Section: 3)û + πF(u ) = Fmentioning

confidence: 99%

A posteriori error analysis for higher order dissipative methods for evolution problems

Makridakis

Nochetto

2006

Numer. Math.

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“…Certain issues as, e.g., a posteriori estimates, estimates of optimal order and regularity, fully discrete schemes with mesh modification, etc., have been extensively considered in the framework of Continuous and Discontinuous Galerkin methods, cf., e.g., [2,[7][8][9][10][11][12][13][14][15][16]. This is probably partly due to the fact that in principle space-time Galerkin methods provide freedom for (almost) arbitrary selection of the space time mesh, [11], and partly due merely to the fact that, as in the elliptic case, the properties of variational methods can be studied in an easier, more systematic and clearer way than properties of their pointwise counterparts, i.e., finite difference methods. Still many issues related to the above problems have to be investigated, mainly for nonlinear evolution PDE's.…”

Section: Introductionmentioning

confidence: 99%

Galerkin time-stepping methods for nonlinear parabolic equations

Akrivis¹,

Makridakis²

2004

ESAIM: M2AN

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Abstract.We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.Mathematics Subject Classification. 65M15, 65M50.

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“…The cG and dG time-stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3,5,10,11,16]. The application of cG and dG approaches to the time discretization of parabolic partial differential equations (PDEs) has been studied in [6][7][8][9]15,28] and the references therein.…”

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confidence: 99%

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

Schötzau

Wihler

2010

Numer. Math.

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The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.

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Galerkin-Type Approximations which are Discontinuous in Time for Parabolic Equations in a Variable Domain

Cited by 142 publications

References 17 publications

A posteriori error analysis for higher order dissipative methods for evolution problems

A posteriori error analysis for higher order dissipative methods for evolution problems

Galerkin time-stepping methods for nonlinear parabolic equations

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

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