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Register a new userUniform preconditioners of linear complexity for problems of negative order
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Added by
Raymond van Veneti\\\"e
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We propose a multi-level type operator that can be used in the framework of operator (or Cald{e}ron) preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions. The cost of applying this multi-level operator scales linearly in the number of mesh cells. Therefore, it provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Math. of Comp., 322(89) (2020), pp. 645-674].
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Using the framework of operator or Cald{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.
In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complement after permuting the original saddle point systems. We discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
Discrete approximations to the equation begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A(x)+H(x)) u^{(1)} + B(x) u = f, ; xin[0,1] end{equation*} are considered. This is an extension of the Sturm-Liouville case $D(x)equiv H(x)equiv 0$ [ M. Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order Sturm-Liouville problems, IMA J. Numer. Anal. {bf 38} (2018), 1485-1522. doi: 10.1093/imanum/drx038] to the non-self-adjoint setting. The natural boundary conditions in the Sturm-Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second, third and fourth order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties--compactness and coercivity. It allows to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.
In this paper we study the linear systems arising from discretized poroelasticity problems. We formulate one block preconditioner for the two-filed Biot model and several preconditioners for the classical three-filed Biot model under the unified relationship framework between well-posedness and preconditioners. By the unified theory, we show all the considered preconditioners are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.
Topology optimization for large scale problems continues to be a computational challenge. Several works exist in the literature to address this topic, and all make use of iterative solvers to handle the linear system arising from the Finite Element Analysis (FEA). However, the preconditioners used in these works vary, and in many cases are notably suboptimal. A handful of works have already demonstrated the effectiveness of Geometric Multigrid (GMG) preconditioners in topology optimization. Here, we show that Algebraic Multigrid (AMG) preconditioners offer superior robustness with only a small overhead cost. The difference is most pronounced when the optimization develops fine-scale structural features or multiple solutions to the same linear system are needed. We thus argue that the expanded use of AMG preconditioners in topology optimization will be essential for the optimization of more complex criteria in large-scale 3D domains.
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