Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

Kopteva, Natalia

doi:10.1090/S0025-5718-2014-02820-2

Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations
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by Natalia Kopteva PDF

Math. Comp. 83 (2014), 2061-2070 Request permission

Abstract:

We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

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