The MINI mixed finite element for the Stokes problem: An experimental investigation

Super-convergence of order 1.5 in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretised with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that super-convergence of order 1.5 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of super-convergence of order 1.5, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

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.Comment: 31 pages, 4 figure

Refined isogeometric analysis for generalized Hermitian eigenproblems

We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku = λMu). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval [λ_s,λ_e] are of interest, we select several shifts σ_k ∈ [λ_s,λ_e] using a spectrum slicing technique. For each shift σ_k, the factorization cost of the spectral transformation matrix K − σ_k M controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K − σ_k M)^−1 M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p−1. When using rIGA, we introduce C^0 separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to O(p^2) in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is O(p). In addition, rIGA improves the accuracy of every eigenpair of the first N_0 eigenvalues and eigenfunctions, where N_0 is the total number of modes of the original maximum-continuity IGA discretization

Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system

The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf-sup stable and pointwise divergence-free. When applied to discretized Stokes equations, these spaces generate a symmetric and indefinite saddle-point linear system. Krylov subspace methods are usually the most efficient procedures to solve such systems. One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES). However, the efficiency and robustness of Krylov subspace methods is closely tied to appropriate preconditioning strategies. For the discrete Stokes system, in particular, block-diagonal strategies provide efficient preconditioners. In this article, we compare the performance of block-diagonal preconditioners for several block choices. We verify how the eigenvalue clustering promoted by the preconditioning strategies affects MINRES convergence. We also compare the number of iterations and wall-clock timings. We conclude that among the building blocks we tested, the strategy with relaxed inner conjugate gradients preconditioned with incomplete Cholesky provided the best results

Refined isogeometric analysis: a solver-based discretization method

112 p.Isogeometric Analysis (IGA) is a computational approach frequently employed nowadaysto study problems governed by partial differential equations (PDEs). This approach definesthe geometry using conventional CAD functions and, in particular, NURBS. Thesefunctions represent complex geometries commonly found in engineering design and arecapable of preserving exactly the geometry description under refinement as required in theanalysis. Moreover, the use of NURBS as basis functions is compatible with theisoparametric concept, allowing to build algebraic systems directly from the computationaldomain representation based on spline functions, which arise from CAD. Therefore, itavoids to define a second space for the numerical analysis resulting in huge reductions inthe total analysis time.For the case of direct solvers, the performance strongly depends upon the employeddiscretization method. In particular, on IGA, the continuity of the solution spaces plays asignificant role in their performance. High continuous spaces degrade the direct solver'sperformance, increasing the solution times by a factor up to O(p^3) with respect totraditional finite element analysis (FEA) per unknown, being p the polynomial order.In this work, we propose a solver-based discretization that employs highly continuous finiteelement spaces interconnected with low continuity hyperplanes to maximize theperformance of direct solvers. Starting from a highly continuous IGA discretization, weintroduce C^0 hyperplanes, which act as separators for the direct solver, to reduce theinterconnection between the degrees of freedom (DoF) in the mesh. By doing so, both thesolution time and best approximation errors are simultaneously improved. We call theresulting method ``refined Isogeometric analysis" (rIGA). Numerical results indicate thatrIGA delivers speed-up factors proportional to p^2. For instance, in a 2D mesh with fourmillion elements and p=5, a Laplace linear system resulting from rIGA is solved 22 timesfaster than the one from highly continuous IGA. In a 3D mesh with one million elementsand p=3, the linear rIGA system is solved 15 times faster than the IGA one.We have also designed and implemented a similar rIGA strategy for iterative solvers. Thisis a hybrid solver strategy that combines a direct solver (static condensation step) toeliminate the internal macro-elements DoF, with an iterative method to solve the skeletonsystem. The hybrid solver strategy achieves moderate savings with respect to IGA whensolving a 2D Poisson problem with a structured mesh and a uniform polynomial degree ofapproximation. For instance, for a mesh with four million elements and polynomial degreep=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGAsystem than to the IGA one. These savings occur because the skeleton rIGA systemcontains fewer non-zero entries than the IGA one. The opposite situation occurs for 3Dproblems, and as a result, 3D rIGA discretizations provide no gains with respect to theirIGA counterparts.Thesis director(s): David Pardo from UPV/EHU university and Victor M. Calo from Curtinuniversit