Numerical approximations for fractional elliptic equations via the method of semigroups.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE(2020)
摘要
We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (-Delta)(s)u = f in omega, subject to some homogeneous boundary conditions BB(u)=0$ \mathcal{B}(u)=0$ on partial differential omega, where s is an element of (0,1), omega subset of Double-struck capital R-n is a bounded domain, and (-Delta)(s) is the spectral fractional Laplacian associated to BB$ \mathcal{B}$ on partial differential omega. We use the solution representation (-Delta)(-s) f together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum f in a suitable fractional Sobolev space of order r >= 0 and the discretization parameter h > 0, our numerical scheme converges as O(h(r+2s)), providing super quadratic convergence rates up to O(h(4)) for sufficiently regular data, or simply O(h(2s)) for merely f is an element of L-2 (omega). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.
更多查看译文
关键词
Fractional Laplacian,bounded domain,boundary value problem,homogeneous and nonhomogeneous boundary conditions,heat semigroup,finite elements,integral quadrature
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络