DeepAI AI Chat
Log In Sign Up

Gibbs Phenomena for L^q-Best Approximation in Finite Element Spaces -- Some Examples

by   Paul Houston, et al.

Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of L^q-best approximation of discontinuities in finite element spaces with 1≤ q<∞. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as q tends to 1. The aim here is to show the potential of L^1 as a solution space in connection with suitably designed meshes.


Constraints for eliminating the Gibbs phenomenon in finite element approximation spaces

One of the major challenges in finite element methods is the mitigation ...

A Construction of C^r Conforming Finite Element Spaces in Any Dimension

This paper proposes a construction of local C^r interpolation spaces and...

There are 174 Triangulations of the Hexahedron

This article answers an important theoretical question: How many differe...

Connections Between Finite Difference and Finite Element Approximations

We present useful connections between the finite difference and the fini...

UFL Dual Spaces, a proposal

This white paper highlights current limitations in the algebraic closure...

Evaluating the Quality of Finite Element Meshes with Machine Learning

This paper addresses the problem of evaluating the quality of finite ele...