A RIESZ BASIS GALERKIN METHOD FOR THE TEMPERED FRACTIONAL LAPLACIAN

摘要

The fractional Laplacian Delta(beta/2) is the generator of the beta-stable Levy process, which is the scaling limit of the Levy flight. Due to the divergence of the second moment of the jump length of the Levy flight, it may not be a suitable physical model in many practical applications. However, using a parameter lambda to exponentially temper the isotropic power law measure of the jump length leads to the tempered Levy flight, which has finite second moment. For a short time the tempered Levy flight exhibits the dynamics of Levy flight, while after a sufficiently long time it turns to normal diffusion. The generator of the tempered beta-stable Levy process is the tempered fractional Laplacian (Delta+lambda)(beta/2) [W. H. Deng et al., Mulbscale Model. Simul., 16 (2018), pp. 125-149]. In the current work, we present new computational methods for the tempered fractional Laplacian equation, including the cases with the homogeneous and nonhomogeneous generalized Dirichlet type boundary conditions. We prove the well-posedness of the Galerkin weak formulation and provide convergence analysis of the single scaling B-spline and multiscale Riesz bases finite element methods. We propose a technique for efficiently generating the entries of the dense stiffness matrix and for solving the resulting algebraic equation by preconditioning. We also present several numerical experiments to verify the theoretical results.