Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis

We consider the stiffness matrices coming from the Galerkin B-spline isogeometric analysis approximation of classical elliptic problems. By exploiting specific spectral properties compactly described by a symbol, we design efficient multigrid methods for the fast solution of the related linear systems. We prove the optimality of the two-grid methods (in the sense that their convergence rate is independent of the matrix size) for spline degrees up to 3, both in the 1D and 2D case. Despite the theoretical optimality, the convergence rate of the two-grid methods with classical stationary smoothers worsens exponentially when the spline degrees increase. With the aid of the symbol, we provide a theoretical explanation of this exponential worsening and by a proper factorization of the symbol we provide a preconditioned conjugate gradient ‘smoother’, in the spirit of the multi-iterative strategy, that allows us to obtain a good convergence rate independent both of the matrix size and of the spline degrees. A selected set of numerical experiments confirms the effectiveness of our proposal and the numerical optimality with a uniformly high convergence rate, also for the V-cycle multigrid method and large spline degrees.