792 research outputs found
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems
We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of Lā(0, T; L2(Ī©)) and the higher order spaces, Lā(0, T;H1(Ī©)) and H1(0, T; L2(Ī©)), with optimal orders of convergence
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model
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