Space-Time adaptive algorithm for the mixed parabolic problem

Abstract

In this paper we present an a-posteriori error estimator for the mixed formulation of a linear parabolic problem, used for designing an efficient adaptive algorithm. Our space-time discretization consists of lowest order Raviart-Thomas finite element over graded meshes and discontinuous Galerkin method with variable time step. Finally, several examples show that the proposed method is efficient and reliable.

A posteriorierror estimates forH1-Galerkin mixed finite-element method for parabolic problems

“…V n h is the Raviart-Thomas interpolation operator [24,25]. Using Cauchy-Schwartz inequality for integrals, and the following approximation properties of Å h [12,24]: for É 2 V, kÉ À Å h Ék L 2 ðKÞ Ch K kr Á Ék L 2 ðKÞ , kr Á ðÉ À Å h ÉÞk L 2 ðKÞ Ckr Á Ék L 2 ðKÞ , we obtain…”

“…For a detailed description of the partition T h,n of and the corresponding finite-element spaces V n h and W n h , we refer to [12,20]. …”

“…For the work on fully discrete a posteriori error analysis by means of the classical mixed method for parabolic problems, we refer to [8,12]. The literature concerning a posteriori error analysis for the standard Galerkin method can be found in [13][14][15][16][17][18][19][20] for linear parabolic problems, and [21,22] for nonlinear parabolic problems.…”

A posteriori error estimates for a fully discrete approximation of Sobolev equations

“…Several approaches have been introduced to define error estimators for parabolic problems (like the heat equation, corresponding to the case where L 1 is reduced to the identity operator), let us quote [4,5,6,7,15,18,21,22,27,28,29]. To be able to extend these techniques to Sobolev equations, we need to be able to manage the replacement of the identity operator by a second order elliptic one.…”

A Posteriori Error Estimation Based on Potential and Flux Reconstruction for the Heat Equation

“…Some other results on a posteriori error estimates for parabolic problems in a conforming setting can also be found in [24] using the so-called functional approach where a flux reconstruction is also considered, but without enforcing any local condition; furthermore, only error upper bounds are derived. Finally, we observe that contrary to conforming finite elements, a posteriori energy-norm error estimates for the heat equation discretized by nonconforming methods are less explored; we mention, in particular, [10] for mixed finite elements, [23] for nonconforming finite elements, [17] for discontinuous Galerkin methods, and [3] for finite volume schemes. This paper is organized as follows.…”