Finite Volume Schemes and Their Improvements for Diffusion Equations with Strong Discontinuities and Anisotropy

Abstract

This paper primarily explores finite volume discretizations of diffusion equations, focusing on the numerical solutions of problems with strong discontinuities and anisotropic coefficients. The research is divided into five parts:

In the first part, a novel gradient transfer method and cascading multigrid method with extrapolation are investigated. This method interpolates vertex unknowns as a linear combination of cell-centered unknowns and employs a cascading multigrid prolongation operator based on the new gradient transfer and splitting extrapolation, forming a cell-centered extrapolation cascading multigrid method suitable for diffusion equations.

The second part delves into the efficient solution of three-dimensional anisotropic diffusion equations. A new vertex unknown interpolation method is introduced, explicitly avoiding the computational cost of solving local linear equation systems. Numerical experiments validate the efficiency and robustness of this method concerning grid size and anisotropic diffusion tensor coefficients.

Following that, an in-depth study is conducted on an unconditionally stable L²-optimal quadratic element finite volume method for solving two-dimensional anisotropic elliptic equations. This new method, based on multiblock control volumes, achieves unconditional stability and L² error convergence of order O(h³) on triangular grids.

The fourth part explores a discrete duality finite volume method with biquadratic elements for solving elliptic equations on quadrilateral meshes. This method exhibits higher accuracy in the discrete L^∞ norm, as confirmed by numerical experiments involving linear and nonlinear elliptic equations with varying parameters and coefficients.

Finally, to address anisotropic elliptic interface problems under non-homogeneous jump conditions, an easily implementable exact-interface-fitted mesh generation algorithm is proposed. A linearity-preserving finite volume scheme is designed for numerical solutions, demonstrating second-order accuracy and linearity-preserving characteristics through numerical validation.

Date of Award	2 Sept 2024
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Weifeng QIU (Supervisor) & Kejia PAN (External Supervisor)