A sparse $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions

TL;DR

This paper introduces an efficient $hp$-finite element method using orthogonal polynomials on circular arcs for solving piecewise-smooth differential equations with periodic boundary conditions, offering advantages over traditional bases especially with discontinuities.

Contribution

The paper develops a novel $hp$-finite element basis on circular arcs that is computationally efficient and well-suited for piecewise-smooth problems with periodic boundary conditions.

Findings

Operators are banded with optimal complexity.

The basis effectively handles discontinuities.

Convergence relates to analyticity regions in the complex plane.

Abstract

We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and achieve optimal complexity regardless of $h$ or $p$, both for building the discretisation and solving the resulting linear system in the case where the operator is symmetric positive definite. The basis serves as a useful alternative to other bases such as the Fourier or integrated Legendre bases, especially for problems with discontinuities. We relate the convergence properties of these bases to regions of analyticity in the complex plane, and further use several differential equation examples to demonstrate these properties. The basis spans the low order eigenfunctions of constant coefficient differential operators, thereby achieving better smoothness properties for time-evolution partial differential equations.