Abstract
We consider the numerical solution of the boundary value problem $$(1) Lu = k(y)u_{xx} + u_{yy} - c(x,y)u = f, u|_{\Gamma _0 \cup \Gamma _1 } = 0,$$ where $$k(y) \gtreqless 0$$ for $$y \gtreqless 0$$ ? 0 and? 1 are parts of the boundary of a bounded simply connected regionG inR 2.G is bounded fory>0 by a piecewise smooth curve? 0 which intersects the liney=0 at the pointsA(?1,0) andB(0, 0). Fory<0,G is bounded by a piecewise smooth curve? 1 throughA, which meets the characteristic of (1) issued fromB at pointC, and by the curve? 2 which consists of the portionCB of the characteristic throughB. Using a weak formulation based on different spaces of test and trial functions, we construct a Galerkin procedure for the above boundary value problem. Existence, uniqueness and uniform stability of an approximate solution is proven and a priori error bounds are given.
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