Work

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

Year: 2025

Type: preprint

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.

Source: arXiv (Cornell University)

Authors: VandenHeuvel, Daniel, Olver, Sheehan

Institution: Imperial College London

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: green

Funders: Imperial College London, Engineering and Physical Sciences Research Council

Award: Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging

OpenAlex ID: W6979234755

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