A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations

2000 ◽  
Vol 53 (5) ◽  
pp. 525-589 ◽  
Author(s):  
Ricardo H. Nochetto ◽  
Giuseppe Savaré ◽  
Claudio Verdi
Keyword(s):  
Error Estimates ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Nonlinear Evolution Equations ◽  
A Posteriori ◽  
Time Step ◽  
A Posteriori Error ◽  
Variable Time Step

scholarly journals A posteriori error estimates based on superconvergence of FEM for fractional evolution equations

2021 ◽  
Vol 19 (1) ◽  
pp. 1210-1222
Author(s):  
Yuelong Tang ◽  
Yuchun Hua
Keyword(s):  
Error Estimates ◽  
Evolution Equations ◽  
Approximation Scheme ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
Triangular Finite Element ◽  
A Posteriori Error ◽  
Convergence And Superconvergence ◽  
Theoretical Results

Abstract In this paper, we consider an approximation scheme for fractional evolution equation with variable coefficient. The space derivative is approximated by triangular finite element and the time fractional derivative is evaluated by the L1 approximation. The main aim of this work is to provide convergence and superconvergence analysis and derive a posteriori error estimates. Some numerical examples are presented to demonstrate our theoretical results.

scholarly journals Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems

2018 ◽  
Vol 39 (3) ◽  
pp. 1158-1179 ◽  
Author(s):  
Alexandre Ern ◽  
Iain Smears ◽  
Martin Vohralík
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Parabolic Problems ◽  
A Posteriori ◽  
Discrete Approximations ◽  
Time Step ◽  
A Posteriori Error ◽  
Norm Estimates ◽  
Norm Of The Error

Abstract We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim \tau $. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq \tau $ or $h^2\simeq \tau $; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition.

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