Upscaling Singular Sources in Weighted Sobolev Spaces by Sub-Grid Corrections

Abstract

In this paper, we develop a numerical multiscale method to solve elliptic boundary value problems with heterogeneous diffusion coefficients and with singular source terms. When the diffusion coefficient is heterogeneous, this adds to the computational costs, and this difficulty is compounded by a singular source term. For singular source terms, the solution does not belong to the Sobolev space $H^1$, but to the space $W^{1,p}$ for some $p<2$. Hence, the problem may be reformulated in a distance-weighted Sobolev space. Using this formulation, we develop a method to upscale the multiscale coefficient near the singular sources by incorporating corrections into the coarse-grid. Using a sub-grid correction method, we correct the basis functions in a distance-weighted Sobolev space and show that these corrections can be truncated to design a computationally efficient scheme with optimal convergence rates. Due to the nature of the formulation in weighted spaces, the variational form must be posed on the cross product of complementary spaces. Thus, two such sub-grid corrections must be computed, one for each multiscale space of the cross product. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Therefore, we develop a weighted projective quasi-interpolation that can be used for a general class of Muckenhoupt weight functions. We verify the optimal convergence of the method in some numerical examples with singular point sources and line fractures, and with oscillatory and heterogeneous diffusion coefficients.