In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in , where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.
In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.In conclusion our numerical benchmarks confirm that the proposed preconditioner is very efficient and well suited for the k-method. Further advances in the solver performance can be achieved with a matrix-free approach, that accelerates the matrix-vector multiplication operation, for moderate or large degree. A first step in this research direction is [30].The outline of the paper is as follows. In Section 2 we give a short review of the Taylor-Hood and Raviart-Thomas isogeometric discretizations for the Stokes system, and summarize the main properties of the Kronecker product. The derivation of the discrete Stokes system is given in Section 3, while in Section 4 we introduce some standard block-structured preconditioners that we will consider in the numerical tests. The core of the paper is Section 5, where we focus on the construction of the preconditioning matrices for the velocity and pressure blocks, discuss their properties and solution strategies. In the Section 6 we propose the modification aimed at improving the preconditioner efficiency by incorporating some information on the geometry parametrization. Numerical results on three different single-patch domains are reported in Section 7. Finally, in Section 8 we draw the conclusions and discuss future directions of research.
Preliminaries
B-splinesIn this section we summarize some basic concepts of B-spline based isogeometric analysis, referring to [2] for the details.Given m and p two positive integers, we introduce a knot vector Ξ := {0 = ξ 1 ≤ ... ≤ ξ m+p+1 = 1} and the associated breakpoint vector Z := {ζ 1 , ..., ζ s }, which contains knots without repetitions. We use open knot vectors, i.e. we suppose ξ 1 = ... = ξ p+1 = 0 and ξ m = ... = ξ m+p+1 = 1 .Then, according to Cox-De Boor recursion formulas [31], we define univariate B-splines as: for p = 0:b 0 α,i (η) =
In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensorproduct construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degreesof-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.A knot vector in [0, 1] is a sequence of non-decreasing points Ξ := {0 = ξ 1 ≤ · · · ≤ ξ m+p+1 = 1}, where m and p are positive integers. We use open knot vectors, that is ξ 1 = · · · = ξ p+1 = 0 and ξ m = · · · = ξ m+p+1 = 1. Then, according to Cox-De Boor recursion formulas (see [14]), the univariate B-splines are piecewise polynomials b i,p : (0, 1) → R defined as for p = 0:
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