Difference schemes for elliptic equations with special boundary conditions

Abstract

On the boundary S of the rectangle a linear combination of the derivative of the unknown function along the inward normal to S, the second derivative along the tangent to the boundary and the values of the unknown function on S are specified. Problems of this type occur, for example, in magnetohydrodynamics [ll in the flow of a conducting fluid with velocity u (x1, x,1 in a magnetic field B in a channel of rectangular section. It is assumed that the walls of the channel are conducting only on segments of finite length. Then we obtain for the potential u in the rectangle G a boundary value problem with boundary conditions of the type indicated above on two opposite sides of the rectangle and u F 0 on the other two sides. In the rectangle itself Au = -f, f = f(v, B). In section 1 we consider the boundary value problem for an elliptic selfconjugate equation which includes the problem of the potential. In section 2 we consider the problem for Poisson’s equation in the rectangle G. On all the sides of the rectangle we specify the boundary conditions