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In this paper, we develop a numerical multiscale method to solve ellipticboundary value problems with heterogeneous diffusion coefficients and withsingular source terms. When the diffusion coefficient is heterogeneous, thisadds to the computational costs, and this difficulty is compounded by asingular source term. For singular source terms, the solution does not belongto 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. Usingthis formulation, we develop a method to upscale the multiscale coefficientnear the singular sources by incorporating corrections into the coarse-grid.Using a sub-grid correction method, we correct the basis functions in adistance-weighted Sobolev space and show that these corrections can betruncated to design a computationally efficient scheme with optimal convergencerates. Due to the nature of the formulation in weighted spaces, the variationalform must be posed on the cross product of complementary spaces. Thus, two suchsub-grid corrections must be computed, one for each multiscale space of thecross product. A key ingredient of this method is the use ofquasi-interpolation operators to construct the fine scale spaces. Therefore, wedevelop a weighted projective quasi-interpolation that can be used for ageneral class of Muckenhoupt weight functions. We verify the optimalconvergence of the method in some numerical examples with singular pointsources and line fractures, and with oscillatory and heterogeneous diffusioncoefficients.

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