Numerical solutions of Dirichlet problem for elliptic operator in divergence form with a right-hand side measure

摘要

We consider a second-order elliptic operator A=A(x)=−∑i,j=1d∂iaij(x)∂j+∑j=1dbj′(x)∂j+∑j=1d∂j(bj″(x)·)+c(x) on Rd from the point of view of its numerical approximations in terms of matrices An having compartmental structure, that is (An)ii>0, (An)ij⩽0, i≠j, ∑i(An)ij⩾0. We solve numerically the corresponding Dirichlet problem on a bounded domain D⊂Rd(d=2,3), for which the right-hand side is a probability measure with support in D. Numerical solutions on grids are nonpositive, and can be naturally embedded into linear spaces of ‘hat’ functions approximating the original solution in Ẇ11(D). Numerical solutions converge in L1(D).The construction of our approximations is valid for general dimensions, but we give the convergence proof only for d=2,3. We end by a nontrivial example that illustrates the obtained results.