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

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• The Helmholtz equation is crucial in wave propagation problems.
• Solving Helmholtz equations with variable coefficients and complex geometries is computationally demanding.
• Existing methods often struggle with discontinuities and anisotropic properties.

Purpose of the Study:

• To develop an efficient and robust numerical method for the Helmholtz equation.
• To handle variable coefficients, radial discontinuities, and anisotropic properties.
• To achieve optimal complexity factorization and quasi-optimal solvers.

Main Methods:

• Sparse hierarchical hp-finite element method (hp-FEM) on disk/annulus domains.
• Preservation of Fourier mode decoupling for rotationally invariant operators.
• Block diagonal mass and stiffness matrices with optimal complexity factorization.
• Handling of radial discontinuities and anisotropic coefficients.

Main Results:

• The hp-FEM yields sparse matrices with a pattern independent of discretization order.
• The method effectively handles radial discontinuities in source terms and coefficients.
• Rotationally anisotropic coefficients approximated by polynomials also lead to sparse systems.
• Quasi-optimal solutions are achieved using the Alternating Direction Implicit (ADI) algorithm.

Conclusions:

• The developed sparse hp-FEM offers an efficient solution for the Helmholtz equation.
• The method demonstrates robustness in handling challenging coefficient variations and discontinuities.
• The approach is extendable to 3D domains and related equations like the Schrödinger equation.