Search

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter., 35 pages, 4 figures, 3 tables

In this note, we provide existence and uniqueness results for frequency domain elastic wave problems. These problems are posed on the complement of a bounded domain Ω⊂R3 (the scatterer). The boundary condition at infinity is given by the Kupradze–Sommerfeld radiation condition and involves different Sommerfeld conditions on different components of the field. Our results are obtained by setting up the problem as a variational problem in the Sobolev space H1 on a bounded domain. We use a nonlocal boundary condition which is related to the Dirichlet to Neumann conditions used for acoustic and electromagnetic scattering problems. We obtain stability results for the source problem, a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences.