Collocation Coarse Approximation (cca) in Multigrid

Abstract. The two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation (DCA) and (Petrov-)Galerkin coarse approximation (GCA). Here, a new approach called collocation coarse approximation (CCA) is introduced, which—like GCA—is algebraically defined and able to cater to difficult features such as discontinuous coefficients, but, unlike GCA, allows explicit control over the coarse-grid sparsity pattern (stencil) and therefore control over the computational complexity of the solver. CCA relies on certain basis functions for which the coarse approximation to the fine-grid problem is exact. Numerical experiments for two-dimensional diffusion problems including jumping coefficients demonstrate the potential of the resulting multigrid algorithms.