%0 Journal Article
%T A Posteriori Error Computations in Finite Element Method for Initial Value Problems
%A K. S. Surana
%A J. Abboud
%J American Journal of Computational Mathematics
%P 81-128
%@ 2161-1211
%D 2025
%I Scientific Research Publishing
%R 10.4236/ajcm.2025.151005
%X A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential: 1) to determine the accuracy of the computed evolution, 2) if the errors in the coupled solutions are higher than an acceptable threshold, then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution. How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution. When <math xmlns='http://www.w3.org/1998/Math/MathML'>
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-norm of the space-time residual over the space-time domain of the computations approaches zero, the approximation <math xmlns='http://www.w3.org/1998/Math/MathML'>
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, the theoretical solution. Thus, the proximity of <math xmlns='http://www.w3.org/1998/Math/MathML'>
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, the <math xmlns='http://www.w3.org/1998/Math/MathML'>
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-norm of the space-time residual function, to zero is a measure of the accuracy or the error in the computed solution. In this paper, we present a methodology and a computational framework for computing <math xmlns='http://www.w3.org/1998/Math/MathML'>
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 in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods. It is shown that the proposed a posteriori computations require <math xmlns='http://www.w3.org/1998/Math/MathML'>
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, <math xmlns='http://www.w3.org/1998/Math/MathML'>
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 framework in both space-time coupled as well as space-time decoupled finite
%K A Posteriori Error Computation
%K Space-Time Coupled
%K Space-Time Decoupled
%K A Priori Error Estimation
%K A Posteriori Error Estimation
%K <i>hpk</i> Scalar Product Spaces
%K Minimally Conforming Scalar Product Spaces
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=141702