Multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers

The maximal regularity properties of a fractional anisotropic elliptic equation are investigated. The equation involves an abstract linear operator A in Fourier type Banach space E and convolution terms. We find the sufficient conditions that guarantee the separability of this problems in E-valued weighted Lp spaces. We prove that an anisotropic elliptic operator generated by this problem is sectorial and generates an analytic semigroup. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation and the boundary value problem for anisotropic elliptic convolution equation are obtained in mixed Lp norms.

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

A full multigrid/full approximation storage (FMG/FAS) method is developed along with a unified pressure-based algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes. Besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system, several flow problems, ranging from incompressible to hypersonic speeds, are computed, in conjunction with an adaptive grid method, to demonstrate its computational capability. With wide variations in geometry, fluid physics, and grid distribution, speedup between the single- and multigrid procedures can be obtained. Because different pressure boundary conditions are needed for incompressible, subsonic, and supersonic inlet and outlet, the number of grid levels that can be effectively used by the present multigrid method appears flow dependent; the implications of this observation are discussed.

In this paper, we study boundary value problems for parameter-dependent elliptic differential-operator equations with variable coefficients in smooth domains. Uniform regularity properties and Fredholmness of this problem are obtained in vector-valued $$ {\text{L}}_{p} $$ -spaces. We prove that the corresponding differential operator is positive and is a generator of an analytic semigroup. Then, via maximal regularity properties of the linear problem, the existence and uniqueness of the solution to the nonlinear elliptic problem is obtained. As an application, we establish maximal regularity properties of the Cauchy problem for abstract parabolic equations, Wentzell–Robin-type mixed problems for parabolic equations, and anisotropic elliptic equations with small parameters.

For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. $$L^p$$, sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.

SummaryMany problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi‐implicit semi‐Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor‐product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor‐product grid with a semi‐structured and quasi‐uniform horizontal mesh and a one‐dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy‐to‐use and scalable environment for algorithms operating on tensor‐product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer. Copyright © 2015 John Wiley & Sons, Ltd.

For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.

We are interested in the design of efficient algebraic multigrid (AMG) methods for the solution of large sparse systems of linear equations arising from finite element (FE) discretization of second-order elliptic partial differential equations (PDEs). In particular, we introduce the concept of so-called “edge matrices”, which–in the present context–are extracted from the individual element matrices. This allows for the construction of spectrally equivalent approximations of the original stiffness matrix that can be utilized in the framework of AMG. The edge matrices give rise to modify the definition of “strong” and “weak” connections (edges), which provides a basis for selecting the coarse-grid nodes in algebraic multigrid methods. Moreover, a reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening algorithms with effective (energy minimizing) interpolation principles involving small-sized “computational molecules” (small collections of edge matrices). This yields a flexible and robust new variant of AMG, which we refer to as AMGm.

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.

The paper is the first part of a program devoted to the study of the behavior of operator-valued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving [Formula: see text]-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisotropic elliptic partial differential equations are established.

The paper deals with the Dirichlet problem for a model nonlinear degenerate anisotropic elliptic second-order equation. Anisotropy and degeneration (with respect to the independent variables) is characterized by the presence of different exponents q1 , q2 and weighted functions |x|^q1 та |x|^q2 in the left side of the equation. The main result of the paper is theorem on the existence of the generalized solution of the Dirichlet problem under consideration.

An efficient finite-difference method for solving Helmholtz equation depends on two points: one is discrete scheme, the other is efficient algorithm. In this paper, we adopt the average-derivative scheme, which owns three advantages : Firstly, it can be applied to unequal directional sampling intervals for Helmholtz equation. Secondly, the scheme is pointwise consistent with Helmholtz equation in a perfect matched layer. And thirdly, it requires less than 4 grid points sampling per wavelength. To solve the discrete Helmholtz equation, which is extremely large and indefinite, direct methods cannot resolve well, and the Krylov subspace iterative methods, such as Bi-CGSTAB and GMRES combining a multigrid-based preconditioner, are good choices. However, the standard multigrid algorithm fails to converge when it encounters unequal directional sampling intervals, which is called anisotropy in multigrid. We analyze the most important three parts of standard multigrid : full weighting restriction operator, point relaxation methods and bilinear interpolation operator, and then we replace them with semi-coarsening, line relaxation and operator-dependent interpolation to make it convergent in anisotropic problems. Consequently, we obtain a satisfactory convergence speed for low and moderate frequency iterative problems in heterogeneous media.

Algebraic multigrid (AMG) methods have proven to be effective for solving the linear algebraic system of equations that arise from many classes of unstructured discretized elliptic PDEs. Standard AMG methods, however, are not suitable for shifted linear systems from elliptic PDEs, such as discretized Helmholtz operators, due to the indefiniteness of the system and the presence of low energy modes which are difficult for multigrid methods to resolve effectively. This paper investigates simple methods to adapt existing standard AMG methods to these shifted systems from direct frequency response analyses in solid mechanics.

A priori estimates for solutions of anisotropic elliptic equations

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations