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

For a second order differential operator $A(\text{msx})=-\mathrm{\nabla }a(\text{msx})\mathrm{\nabla }+b^{\mathrm{\prime }}(\text{msx})\mathrm{\nabla }+\mathrm{\nabla }({\text{msb}}^{\mathrm{\prime }\mathrm{\prime }}(\text{msx})\cdot )$ on a bounded domain $D$ with the Dirichlet boundary conditions on $\mathrm{\partial }D$ there exists the inverse $T(\lambda ,A)=(\lambda I+A{)}^{-1}$ in $L_1(D)$. If $\mu$ is a Radon (probability) measure on Borel algebra of subsets of $D$, then $T(\lambda ,A)\mu \in L_p(D),p\in [1,d\mathrm{/}(d-1))$. We construct the numerical approximations to $u=T(\lambda ,A)\mu$ in two steps. In the first one we construct grid-solutions ${\mathbf{u}}_n$ 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 ${\dot{W}}_p^1(D)$.