A Sparse Hierarchical hp-Finite Element Method on Disks and Annuli

• Published in: Journal of scientific computing Vol. 104; no. 2; p. 51
• Main Authors: Papadopoulos, Ioannis P. A., Olver, Sheehan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2025; Springer Nature B.V
• Subjects: Algorithms; Annuli; Cartesian coordinates; Computational Mathematics and Numerical Analysis; Decoupling; Discontinuity; Discretization; Disks; Finite element analysis; Finite element method; Helmholtz equations; Invariants; Linear systems; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Partial differential equations; Polynomials; Schrodinger equation; Sparsity; Stiffness matrix; Theoretical; Cylinder; Quasi-optimal complexity; finite element method; High-frequency Helmholtz equation; Annulus; Disk; Schrödinger equation; hp-finite element method
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-025-02964-4

We develop a sparse hierarchical hp -finite element method ( hp -FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse hp -FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. e consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, țhe time-dependent Schrödinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm.