Numerical approach to $L_1$-problems with the second order elliptic operators

Abstract

For a second order differential operator $A(\msx) =-\nabla a(\msx)\nabla + b'(\msx)\nabla+ \nabla \big(\msb''(\msx) \cdot\big)$ on a bounded domain $D$ with the Dirichlet boundary conditions on $\partial D$ there exists the inverse $T(λ, A)= (λI+A)^{-1}$ in $L_1(D)$. If $μ$ is a Radon (probability) measure on Borel algebra of subsets of $D$, then $T(λ, A)μ\in L_p(D), p \in [1, d/(d-1))$. We construct the numerical approximations to $u =T(λ, A)μ$ in two steps. In the first one we construct grid-solutions ${\bf u}_n$ and in the second step we embed grid-solutions into the linear space of hat functions $u(n) \in \dot{W}_p^1(D)$. The strong convergence to the original solutions $u$ is established in $L_p(D)$ and the weak convergence in $\dot{W}_p^1(D)$.