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The extraction of substrate coupling resistances can be formulated as a first-kind integral equation, which requires only discretization of the two-dimensional contacts. However, the result is a dense matrix problem which is too expensive to store or to factor directly. Instead, we present a novel, multigrid iterative method which converges more rapidly than previously applied Krylov-subspace methods. At each level in the multigrid hierarchy, we avoid dense matrix-vector multiplication by using moment-matching approximations and a sparsification algorithm based on eigendecomposition. Results on realistic examples demonstrate that the combined approach is up to an order of magnitude faster than a Krylov-subspace method with sparsification, and orders of magnitude faster than not using sparsification at all.