Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

Abstract

In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9-point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9-point compact stencil, no special formulas are needed near the boundaries. Both second-order and fourth-order discretizations are derived.The fourth-order approximations produce more accurate results than the 13-point classical stencil or the commonly used system of two second-order equations coupled with the boundary condition.The method suffers from slow convergence when classical iteration methods such as Gauss-Seidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques that exhibit grid-independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from three different problems, including Stokes flow in a driven cavity, are reported.

Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries

“…Shankar 30 also studied the cavity flow, though he considered only the motion of one endwall and used a numerical method using a truncated eigenfunction expansion. Here we show solutions obtained by a finite difference scheme developed by Altas et al 31 This scheme uses not only the streamfunction as unknown but also the first derivatives. This expands the number of unknowns to three.…”

A Compact Scheme for the Streamfunction Formulation of Navier-Stokes Equations

“…al. [1] and Kupferman [5] applied Stephenson's scheme, using a multigrid solver. Stephenson's compact approximation for the biharmonic operator is the following.…”

A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains

“…Due to the significance of the biharmonic operator a large number of methods for discretizing (1) have been proposed. It seems that the majority of these methods are related to the finite elements methodology (see for example [3,10,14,15,29] and references therein).…”

“…This method does not require any modifications near the boundary, as the boundary conditions also include the values of ∇Φ [31]. Some variations and a multigrid technique were proposed in [1]. So far, all the algorithms based on the nine-point stencil were only applicable to a rectangular domain.…”