Volume 28, Issue 1
Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations

J. Comp. Math., 28 (2010), pp. 1-10.

Published online: 2010-02

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• Abstract

We construct and analyse a nodal $\mathcal{O}(h^4)$-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal $\mathcal{O}(h^4)$-superconvergence (ultraconvergence). The obtained superconvergence result is illustrated by two numerical examples.

• Keywords

Higher order error estimates, Tetrahedral and prismatic elements, Superconvergence, Averaging operators.

65N30.

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@Article{JCM-28-1, author = {}, title = {Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {1}, pages = {1--10}, abstract = {

We construct and analyse a nodal $\mathcal{O}(h^4)$-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal $\mathcal{O}(h^4)$-superconvergence (ultraconvergence). The obtained superconvergence result is illustrated by two numerical examples.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m1004}, url = {http://global-sci.org/intro/article_detail/jcm/8503.html} }
TY - JOUR T1 - Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 10 PY - 2010 DA - 2010/02 SN - 28 DO - http://doi.org/10.4208/jcm.2009.09-m1004 UR - https://global-sci.org/intro/article_detail/jcm/8503.html KW - Higher order error estimates, Tetrahedral and prismatic elements, Superconvergence, Averaging operators. AB -

We construct and analyse a nodal $\mathcal{O}(h^4)$-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal $\mathcal{O}(h^4)$-superconvergence (ultraconvergence). The obtained superconvergence result is illustrated by two numerical examples.

Antti Hannukainen, Sergey Korotov & Michal Křížek. (2019). Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations. Journal of Computational Mathematics. 28 (1). 1-10. doi:10.4208/jcm.2009.09-m1004
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