An efficient multigrid solver for 3D biharmonic equation with a discretization by 25-point difference scheme

TL;DR

The paper tackles solving the 3D biharmonic equation $\Delta^2 u=f$ on $\Omega=[0,1]^3$ with Dirichlet boundary conditions, a challenging large-scale non-symmetric linear system arising from a 25-point FD discretization. It introduces the EXCMG_bi-cg framework, which uses Richardson extrapolation and quadratic interpolation to generate a third-order accurate initial guess on progressively finer grids and solves the resulting systems with Bi-CG under a relative residual tolerance. The main contributions are the EXCMG algorithm, its 3D extension with extrapolation on embedded meshes, and extensive numerical demonstrations up to about $135$ million unknowns showing dramatic reductions in iterations and total work. The approach provides a scalable, high-accuracy solver for complex 3D biharmonic problems with potential applications in elasticity and related fields, leveraging explicit multigrid extrapolation techniques with a robust iterative solver.

Abstract

In this paper, we propose an efficient extrapolation cascadic multigrid (EXCMG) method combined with 25-point difference approximation to solve the three-dimensional biharmonic equation. First, through applying Richardson extrapolation and quadratic interpolation on numerical solutions on current and previous grids, a third-order approximation to the finite difference solution can be obtained and used as the iterative initial guess on the next finer grid. Then we adopt the bi-conjugate gradient (Bi-CG) method to solve the large linear system resulting from the 25-point difference approximation. In addition, an extrapolation method based on midpoint extrapolation formula is used to achieve higher-order accuracy on the entire finest grid. Finally, some numerical experiments are performed to show that the EXCMG method is an efficient solver for the 3D biharmonic equation.

Figures (3)

• Figure 1: The four level structure of the V- and W-cycles, CMG and EXCMG methods. In the diagram, $\bullet$ denotes pre-smoothing steps, $\circ$ denotes post-smoothing steps, $\uparrow$ denotes prolongation, $\downarrow$ denotes restriction, $\Uparrow$ denotes extrapolation and quadratic interpolation, and $\blacksquare$ denotes direct solver.
• Figure 2: Three embedded mesh in 1D.
• Figure 3: Three embedded hexahedral mesh

Theorems & Definitions (1)

• Remark 1