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In this paper, we introduce the isoGeometric Residual Minimization (iGRM)method. The method solves stationary advection-dominated diffusion problems. Westabilize the method via residual minimization. We discretize the problem usingB-spline basis functions. We then seek to minimize the isogeometric residualover a spline space built on a tensor product mesh. We construct the solutionover a smooth subspace of the residual. We can specify the solution subspace byreducing the polynomial order, by increasing the continuity, or by acombination of these. The Gramm matrix for the residual minimization method isapproximated by a weighted H1 norm, which we can express as Kronecker products,due to the tensor-product structure of the approximations. We use the Grammmatrix as a preconditional which can be applied in a computational costproportional to the number of degrees of freedom in 2D and 3D. Building onthese approximations, we construct an iterative algorithm. We test the residualminimization method on several numerical examples, and we compare it to theDiscontinuous Petrov-Galerkin (DPG) and the Streamline Upwind Petrov-Galerkin(SUPG) stabilization methods. The iGRM method delivers similar qualitysolutions as the DPG method, it uses smaller grids, it does not requirebreaking of the spaces, but it is limited to tensor-product meshes. Thecomputational cost of the iGRM is higher than for SUPG, but it does not requirethe determination of problem specific parameters.

Isogeometric Residual Minimization Method (iGRM) with Direction Splitting Preconditoner for Stationary Advection-Diffusion Problems