Numerische Mathematik

Numer Math 28,243-258(1977)

Numerische Mathematik

© by Springer-Verlag 1977

Composite Mesh Difference Methods for Elliptic Boundary Value Problems*

Goran Starius

Department of Computer Science, Uppsala University, Uppsala, Sweden

Summary . The purpose of this paper is to develop composite mesh difference methods for eUiptic boundary value problems over regions with curved, smooth boundaries. A curved mesh will cover an annular strip along the boundary of the region which is included in the mesh. For the rest of the region and for a suitable inner part of the annular strip a square or rectangular mesh will be used. On each mesh a difference approximation is set up as well as couplings between them. Only second order methods for second order elliptic equations will be treated in detail.

1 . Introduction

Let the boundary ôQ of a region Q in an xy-plane be a simple smooth curve. A rectangle jR = {(s,r)|0^s^S,0^r^Z^-^} in an sr-plane is mapped onto an annular subregion Q2 of ß by a smooth transformation T{s, r) which is periodic in s with period S. The outer boundary of Q2 is dQ, which is given by s^T{s,0). A square mesh in R is mapped onto an orthogonal curvilinear mesh in 02- F^^* the determination of the transformation Tsee [5]. A square or rectangular mesh covers a subregion of Q corresponding to the interior of a curve s-> T{s, /^\ 0</^^< /^\ The inner boundary of Q2 is s-^Tis^r^^-^). The elliptic equation together with the boundary conditions are transformed to the rectangle R. Now the differential equation is discretized on the rectilinear mesh in Q and on R, where also the boundary conditions are replaced by suitable difference approximations. The two schemes are coupled by using central interpolation formulas at the inner boundary of the curvilinear mesh and at the boundary of the rectilinear one. This means that the interior scheme will always be a Dirichlet problem. The main advantages of this technique over that of using only a rectilinear mesh are that derivatives in boundary conditions can easily be approximated, extrapolation or uncentered interpolation formulas can be avoided at the boundary, for compact schemes, and that the stability problem is made easier.

* This research was supported by the Swedish Institute for Applied Mathematics (ITM)