Name
Abstract | ||
|---|---|---|
In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form; however, the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a subgrid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is our main interest, we construct a projective quasi-interpolation that has both d and d + 1 dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, utilizing a global L-2 projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension 2 + 1 for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1137/17M1147305 | MULTISCALE MODELING & SIMULATION |
Keywords | Field | DocType |
localization,multiscale methods,fractional Laplacian,heterogeneous diffusion,nonlocal models | Convergence (routing),Degenerate energy levels,Homogenization (chemistry),Mathematical analysis,Sobolev space,Fractional Laplacian,Basis function,Operator (computer programming),Rate of convergence,Mathematics | Journal |
Volume | Issue | ISSN |
16 | 3 | 1540-3459 |
Citations | PageRank | References |
0 | 0.34 | 8 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Donald L. Brown | 1 | 22 | 3.63 |
| Joscha Gedicke | 2 | 58 | 9.24 |
| Daniel Peterseim | 3 | 104 | 13.03 |