Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique

TL;DR

This paper introduces a new polynomial algebra-based finite element method for curved domains that avoids the complexities of isoparametric techniques, maintaining optimal approximation properties without curved elements.

Contribution

It proposes a simple, algebraic alternative to isoparametric methods for finite element approximation in curved domains, applicable to arbitrary order and avoiding curved elements.

Findings

Achieves optimal approximation properties in curved domains

Eliminates the need for curved elements in finite element meshes

Demonstrates advantages through numerical examples

Abstract

Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of the finite-element approach to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of function-value Dirichlet conditions prescribed on curvilinear boundaries method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard straight-edged elements in the case of polygonal or polyhedral domains. However, besides obvious algebraic and geometric inconveniences, the isoparametric technique is helplessly limited in scope and simplicity, since its extension to degrees of freedom other than function values is not straightforward if not unknown. The purpose of this paper is to propose, study and test a simple alternative that bypasses all the above drawbacks, without eroding qualitative approximation properties. More specifically this technique can do without curved elements and is based only on polynomial algebra. REMARKS : First submission (dated Jan. 3, 2017) updated on Jan. 11, 2017 with the addition of a footnote on author's research grant. A third version with several improvements was posted on Nov. 2, 2017. The fourth version incorporated some findings during the revision of a related paper. In the fifth version, besides minor changes, the convection-diffusion equation became the model problem; the text was reviewed in order to highlight the advantages of the new approach over classical techniques, more particularly by means of additional numerical examples.