scholarly journals Interior a posteriori error estimates for time discrete approximations of parabolic problems

2013 ◽  
Vol 124 (3) ◽  
pp. 541-557 ◽  
Author(s):  
Christian Lubich ◽  
Charalambos Makridakis
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Parabolic Problems ◽  
A Posteriori ◽  
Discrete Approximations ◽  
A Posteriori Error

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.

scholarly journals A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems

2009 ◽  
Vol 47 (3) ◽  
pp. 2157-2176 ◽  
Author(s):  
Alan Demlow ◽  
Omar Lakkis ◽  
Charalambos Makridakis
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Maximum Norm ◽  
Parabolic Problems ◽  
A Posteriori ◽  
A Posteriori Error
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