Abstract
In this paper we introduce the semiregularity property for a family of decompositions of a polyhedron into a natural class of prisms. In such a family, prismatic elements are allowed to be very flat or very long compared to their triangular bases, and the edges of quadrilateral faces can be nonparallel. Moreover, the triangular faces of each element are either parallel or skew to each other. To estimate the error of the interpolation operator defined on the finite space whose basis functions are defined on the general prismatic elements, we consider quadratic polynomials as the basis functions for that space which are bilinear on the reference prism. We then prove that under this modification of the semiregularity criterion, the interpolation error is of order O(h) in the H1-norm.
Highlights
The finite element method is one of the most flexible and powerful methods to solve numerically a wide variety of partial differential equations [3,13,11,14]
A fundamental problem is to estimate the error between the exact solution and its computable finite element approximation
For linear elliptic boundary value problems in 2-dimensional space, Zlámal [15] introduced the minimum angle condition that guarantees a bound on the constant in the final error which comes from the estimation error of the defined interpolation operator
Summary
The finite element method is one of the most flexible and powerful methods to solve numerically a wide variety of partial differential equations [3,13,11,14]. The maximum angle condition enables us to keep an optimal error whereas we are allowed to consider degenerating families of elements in order to cover the narrow or flat parts of a given bounded domain. The aim of this paper is to estimate the interpolation error for a more general class of prismatic elements than previously considered in [6]. This class of elements naturally appear e.g. in some standard geometric models.
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