Optimal-rate Lagrange and Hermite finite elements for Dirichlet problems in curved domains with straight-edged triangles

TL;DR

This paper introduces an alternative finite element approach for Dirichlet problems in curved domains that avoids curved elements and relies solely on polynomial algebra, maintaining optimal approximation properties.

Contribution

It proposes a simple, algebraic method for finite element approximation in curved domains that bypasses the limitations of isoparametric techniques, applicable to Lagrange and Hermite elements.

Findings

Achieves optimal approximation without curved elements

Applicable to convection-diffusion and biharmonic equations

Maintains qualitative approximation properties

Abstract

One of the reasons for the success of the finite element method is its versatility to deal with different types of geometries. This is particularly true of problems posed in curved domains of arbitrary shape. In the case of second order boundary-value problems with 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 methods of order greater than one based on standard straight-edged elements, in the case of polygonal or polyhedral domains. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. The purpose of this paper is to study a simple alternative that bypasses the above drawbacks, without eroding qualitative approximation properties. Among other advantages, this technique can do without curved elements and is based only on polynomial algebra. It is first illustrated in the case of the convection-diffusion equation solved with standard Lagrange elements. Then it is applied to the solution with Hermite elements of the biharmonic equation with Dirichlet boundary conditions.