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

## TL;DR

This paper introduces an efficient multigrid solver combining extrapolation techniques and a 25-point difference scheme to solve the 3D biharmonic equation with high accuracy and computational efficiency.

## Contribution

The paper develops a novel extrapolation cascadic multigrid method with 25-point discretization and demonstrates its effectiveness for 3D biharmonic problems.

## Key findings

- The method achieves higher-order accuracy on the finest grid.
- Numerical experiments confirm the efficiency and accuracy of the solver.
- The approach outperforms traditional methods in solving 3D biharmonic equations.

## 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.