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

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

2007-12-21

arxiv.org/abs/0712.3678v2

Mathematics

Analysis of PDEs

33 pages

Scientific paper

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 $T(\lambda, A)= (\lambda I+A)^{-1}$ in $L_1(D)$. If $\mu$ is a Radon (probability) measure on Borel algebra of subsets of $D$, then $T(\lambda, A)\mu \in L_p(D), p \in [1, d/(d-1))$. We construct the numerical approximations to $u =T(\lambda, A)\mu$ 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 $\dot{W}_p^1(D)$.

Affiliated with

Also associated with

No associations

Rating

Numerical approach to $L_1$-problems with the second order elliptic operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Numerical approach to $L_1$-problems with the second order elliptic operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Numerical approach to $L_1$-problems with the second order elliptic operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-696671