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Volume 2, Issue 3
Control of Geometry Induced Error in $hp$ Finite Element (FE) Simulations, I. Evaluation of FE Error for Curvilinear Geometries

D. Xue & L. Demkowicz

Int. J. Numer. Anal. Mod., 2 (2005), pp. 283-300.

Published online: 2005-02

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  • Abstract

The paper discusses a general framework for handling curvilinear geometries in high accuracy Finite Element (FE) simulations, for both elliptic and Maxwell problems. Based on the differential manifold concept, the domain is represented as a union of geometrical blocks prescribed with globally compatible, explicit or implicit parameterizations. The idea of parametric $H^1$-, $H(curl)$- and $H(div)$-conforming elements is reviewed, and the concepts of exact geometry elements and isoparametric elements are discussed. The paper focuses then on isoparametric elements, and two ways of computing FE discretization errors: a popular one, neglecting the geometry approximation, and a precise one, utilizing the exact geometry representation. Presented numerical examples indicate the necessity of accounting for the geometry error in FE error calculations, especially for the $H(curl)$ problems.

  • Keywords

geometry approximation, curvilinear $hp$ Finite Element(FE) meshes, error evaluation, Exact Geometry Integration (EGI).

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-2-283, author = {D. Xue and L. Demkowicz}, title = {Control of Geometry Induced Error in $hp$ Finite Element (FE) Simulations, I. Evaluation of FE Error for Curvilinear Geometries}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {3}, pages = {283--300}, abstract = {

The paper discusses a general framework for handling curvilinear geometries in high accuracy Finite Element (FE) simulations, for both elliptic and Maxwell problems. Based on the differential manifold concept, the domain is represented as a union of geometrical blocks prescribed with globally compatible, explicit or implicit parameterizations. The idea of parametric $H^1$-, $H(curl)$- and $H(div)$-conforming elements is reviewed, and the concepts of exact geometry elements and isoparametric elements are discussed. The paper focuses then on isoparametric elements, and two ways of computing FE discretization errors: a popular one, neglecting the geometry approximation, and a precise one, utilizing the exact geometry representation. Presented numerical examples indicate the necessity of accounting for the geometry error in FE error calculations, especially for the $H(curl)$ problems.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/932.html} }
TY - JOUR T1 - Control of Geometry Induced Error in $hp$ Finite Element (FE) Simulations, I. Evaluation of FE Error for Curvilinear Geometries AU - D. Xue & L. Demkowicz JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 283 EP - 300 PY - 2005 DA - 2005/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/932.html KW - geometry approximation, curvilinear $hp$ Finite Element(FE) meshes, error evaluation, Exact Geometry Integration (EGI). AB -

The paper discusses a general framework for handling curvilinear geometries in high accuracy Finite Element (FE) simulations, for both elliptic and Maxwell problems. Based on the differential manifold concept, the domain is represented as a union of geometrical blocks prescribed with globally compatible, explicit or implicit parameterizations. The idea of parametric $H^1$-, $H(curl)$- and $H(div)$-conforming elements is reviewed, and the concepts of exact geometry elements and isoparametric elements are discussed. The paper focuses then on isoparametric elements, and two ways of computing FE discretization errors: a popular one, neglecting the geometry approximation, and a precise one, utilizing the exact geometry representation. Presented numerical examples indicate the necessity of accounting for the geometry error in FE error calculations, especially for the $H(curl)$ problems.

D. Xue and L. Demkowicz. (2005). Control of Geometry Induced Error in $hp$ Finite Element (FE) Simulations, I. Evaluation of FE Error for Curvilinear Geometries. International Journal of Numerical Analysis and Modeling. 2 (3). 283-300. doi:
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