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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} }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.