Abstract
The swift, improvement of computational capabilities enables us to apply finite element methods to simulate more and more problems arising from various applications. A fundamental question associated with finite element simulations is their accuracy. In other words, before we can make any decisions based on the numerical solutions, we must be sure that they are acceptable in the sense that their errors are within the given tolerances. Various estimators have been developed to assess the accuracy of finite element solutions, and they can be classified basically into two types: a priori error estimates and a posteriori error estimates. While a priori error estimates can give us asymptotic convergence rates of numerical solutions in terms of the grid size before the computations, they depend on certain Sobolev norms of the true solutions which are not known, in general. Therefore, it is difficult, if not impossible, to use a priori estimates directly to decide whether a numerical solution is acceptable or a finer partition (and so a new numerical solution) is needed. In contrast, a posteriori error estimates depends only on the numerical solutions, and they usually give computable quantities about the accuracy of the numerical solutions.
This research was supported in part by National Science Foundation Grants Nos. DMS-8922865 and EPSCoR EHR-9108772, by funding from DOE, DE-ACO5-840R21400, Martin Marietta, Subcontract, SK965C and SK966V, by funding from the Institutes for Scientific Computation at Texas A&M University, and by funding from Office of Naval Research Contract No. 0014-88-K-0370.
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Lin, T., Wang, H. (1995). A Class of Error Estimators Based on Interpolating the Finite Element Solutions for Reaction-Diffusion Equations. In: Babuska, I., Henshaw, W.D., Oliger, J.E., Flaherty, J.E., Hopcroft, J.E., Tezduyar, T. (eds) Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4248-2_7
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