An approach is given for solving large linear systems that combines Krylov methods with use of two different grid levels. Eigenvectors are computed on the coarse grid and used to deflate eigenvalues on the fine grid. GMRES-type methods are first used on both the coarse and fine grids. Then another approach is given that has a restarted BiCGStab (or IDR) method on the fine grid. While BiCGStab is generally considered to be a non-restarted method, it works well in this context with deflating and restarting. Tests show this new approach can be very efficient for difficult linear equations problems.
Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer
We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs. The method involves two stages, setup and solve. In the setup stage, hierarchical coarse grids are constructed through aggregation of the fine grid nodes. These aggregations are obtained using a set of maximal independent nodes from the fine grid nodes. We use a ``fine-grain'' parallel algorithm for finding a maximal independent set from a graph of strong ne…
A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known t…
Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and sometimes ill-conditioned operators. We review common discretization methods for elliptic PDEs such as the finite difference, finite volume, finite element, and spectral methods and the linear systems they form. We also provide an overview of classic to modern solution methods for the linear systems formed by these discretization methods. These methods include s…
AlgMortar: a fully algebraic multiscale mortar preconditioner
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo（圣保罗大学数学与计算科学学院）；Piri Technologies, LLC（Piri科技有限公司）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学科学系）
The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary integral (KFBI) method which works well on complex domains to efficiently simulate the linear diffusion part of the bidomain equation. After a proper temporal discretization, the KFBI method is applied to solve the resulting homogeneous Neumann boundary value problems with a second-order accuracy. According to the potential theory, the boundary integral equations …
