Numerical Approximation of the Integral Fractional Laplacian
This work addresses the computational challenge of solving fractional Laplacian problems, which are important in anomalous diffusion and nonlocal phenomena, but the results are incremental as they combine existing techniques without a breakthrough in efficiency or accuracy.
The paper proposes a new nonconforming finite element algorithm for solving elliptic problems involving the fractional Laplacian, using an integral representation combined with sinc quadrature, domain truncation, and finite element approximation. Numerical results demonstrate error behavior in terms of mesh size, truncation parameter, and quadrature spacing.
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.