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Register a new userGeneral theory of interpolation error estimates on anisotropic meshes, part II
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Added by
Hiroki Ishizaka
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a general theory of interpolation error estimates for smooth functions and inverse inequalities on anisotropic meshes. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. This paper also includes corrections to an error in General theory of interpolation error estimates on anisotropic meshes (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 2 was incorrect.
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We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.
In the error analysis of finite element methods, the shape regularity assumption on triangulations is typically imposed to obtain a priori error estimations. In practical computations, however, very thin or degenerated elements that violate the shape regularity assumption may appear when we use adaptive mesh refinement. In this manuscript, we attempt to establish an error analysis approach without the shape regularity assumption on triangulations. We have presented several papers on the error analysis of finite element methods on non-shape regular triangulations. The main points in these papers are that, in the error estimates of finite element methods, the circumradius of the triangles is one of the most important factors. The purpose of this manuscript is to provide a simple and plain explanation of the results to researchers and, in particular, graduate students who are interested in the subject. Therefore, this manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.
This is the second lecture note on the error analysis of interpolation on simplicial elements without the shape regularity assumption (the previous one is arXiv:1908.03894). In this manuscript, we explain the error analysis of Lagrange interpolation on (possibly anisotropic) tetrahedrons. The manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.
This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.
Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.
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