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

Abstract

We consider a second-order elliptic operator A=A( x )=−∑ i,j=1 d ∂ ia ij( x ) ∂ j+∑ j=1 db j′( x ) ∂ j+∑ j=1 d ∂ j(b j″( x )·)+c( x ) on R d from the point of view of its numerical approximations in terms of matrices A n having compartmental structure, that is ( A n ) ii >0, ( A n ) ij ⩽0, i≠ j, ∑ i ( A n ) ij ⩾0. We solve numerically the corresponding Dirichlet problem on a bounded domain D⊂ R d(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 W ̇ 1 1(D) . Numerical solutions converge in L 1( 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.