Nonconforming h-p spectral element methods for elliptic problems

TL;DR

This paper presents a modified nonconforming h-p spectral element method for solving elliptic problems on polygonal domains with exponential accuracy, leveraging specialized meshes and stability estimates.

Contribution

It introduces a faster, more accurate nonconforming spectral element method with improved error estimates for elliptic problems on polygonal domains.

Findings

Achieves exponential accuracy on polygonal domains.

Provides stability estimates based on regularity results.

Demonstrates improved computational speed over previous methods.

Abstract

In this paper we show that we can use a modified version of the h-p spectral element method proposed in \cite{duttora1,duttom,duttora2,tomarth} to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in \cite{babguo1}. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in \cite{duttora1,duttom,tomarth} and the h-p finite element method and stronger error estimates are obtained.