skip to main content

Search for: All records

Award ID contains: 2012291

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. Abstract

    It is known that the solutions to space‐fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill‐conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two‐sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three‐by‐three block linear system are proved to be positive‐definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner31as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill‐conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent.

    more » « less
  2. Free, publicly-accessible full text available October 1, 2024
  3. Free, publicly-accessible full text available August 18, 2024
  4. Free, publicly-accessible full text available June 15, 2024
  5. Free, publicly-accessible full text available June 1, 2024
  6. Free, publicly-accessible full text available May 15, 2024
  7. Free, publicly-accessible full text available May 1, 2024
  8. We prove the well‐posedness and smoothing properties of a distributed‐order time‐fractional diffusion equation with a singular density function in multiple space dimensions, which could model the ultraslow subdiffusion processes. We accordingly derive a finite element approximation to the problem and prove its optimal‐order error estimate. Numerical results are presented to support the mathematical and numerical analysis.

    more » « less