Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. (English) Zbl 1102.65125
Summary: We propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of the second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss-Seidel and successive overrelaxation for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use the multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions.
MSC:
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
multigrid method; finite difference schemes; three-dimensional biharmonic equation; high accuracy; compact approximations; numerical examples; Gauss-Seidel method; comparison of methods; convergence; successive overrelaxationSoftware:
WesselingReferences:
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