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Register a new userAlgebraic Multilevel Preconditioning in Isogeometric Analysis: Construction and Numerical Studies
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We present algebraic multilevel iteration (AMLI) methods for isogeometric discretization of scalar second order elliptic problems. The construction of coarse grid operators and hierarchical complementary operators are given. Moreover, for a uniform mesh on a unit interval, the explicit representation of B-spline basis functions for a fixed mesh size $h$ is given for $p=2,3,4$ and for $C^{0}$- and $C^{p-1}$-continuity. The presented methods show $h$- and (almost) $p$-independent convergence rates. Supporting numerical results for convergence factor and iterations count for AMLI cycles ($V$-, linear $W$-, nonlinear $W$-) are provided. Numerical tests are performed, in two-dimensions on square domain and quarter annulus, and in three-dimensions on quarter thick ring.
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An algebraic multilevel iteration method for solving system of linear algebraic equations arising in $H(mathrm{curl})$ and $H(mathrm{div})$ spaces are presented. The algorithm is developed for the discrete problem obtained by using the space of lowest order Nedelec and Raviart-Thomas-Nedelec elements. The theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local (element-wise) problems. In the hierarchical basis framework, explicit recursion formulae are derived to compute the element matrices and the constant $gamma$ (which measures the quality of the space splitting) at any given level. It is proved that the proposed method is robust with respect to the problem parameters, and is of optimal order complexity. Supporting numerical results, including the case when the parameters have jumps, are also presented.
This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.
The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order $p$. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as $p^{d+2}$ instead of $p^{2d+1}$ as previously achieved.
The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical algebraic multigrid is applied to solve the subsystems that arise in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We use representative one-group and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, and scale well both algorithmically and in parallel.
In this article, we discuss the numerical solution of diffusion equations on random surfaces within the isogeometric framework. Complex computational geometries, given only by surface triangulations, are recast into the isogeometric context by transforming them into quadrangulations and a subsequent interpolation procedure. Moreover, we describe in detail, how diffusion problems on random surfaces can be modelled and how quantities of interest may be derived. In particular, we propose a low rank approximation algorithm for the high-dimensional space-time correlation of the random solution. Extensive numerical studies are provided to quantify and validate the approach.
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