505 research outputs found
Block-adaptive Cross Approximation of Discrete Integral Operators
In this article we extend the adaptive cross approximation (ACA) method known
for the efficient approximation of discretisations of integral operators to a
block-adaptive version. While ACA is usually employed to assemble hierarchical
matrix approximations having the same prescribed accuracy on all blocks of the
partition, for the solution of linear systems it may be more efficient to adapt
the accuracy of each block to the actual error of the solution as some blocks
may be more important for the solution error than others. To this end, error
estimation techniques known from adaptive mesh refinement are applied to
automatically improve the block-wise matrix approximation. This allows to
interlace the assembling of the coefficient matrix with the iterative solution
Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers
In this article, we focus on a two-dimensional conservative steady-state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)-based discretization approach. Precisely, we use both classical FVs and the so-called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first-order discretizations. By exploiting the Toeplitz-like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second-order FV discretization. This same information is leveraged to discuss parameter-free symbol-based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well
HAZniCS -- Software Components for Multiphysics Problems
We introduce the software toolbox HAZniCS for solving interface-coupled
multiphysics problems. HAZniCS is a suite of modules that combines the
well-known FEniCS framework for finite element discretization with solver and
graph library HAZmath. The focus of the paper is on the design and
implementation of a pool of robust and efficient solver algorithms which tackle
issues related to the complex interfacial coupling of the physical problems
often encountered in applications in brain biomechanics. The robustness and
efficiency of the numerical algorithms and methods is shown in several
numerical examples, namely the Darcy-Stokes equations that model flow of
cerebrospinal fluid in the human brain and the mixed-dimensional model of
electrodiffusion in the brain tissue
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