The main purpose of this paper is to give a systematic introduction number iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works methods, unified theory diverse group algorithms, such as Jacobi Gauss–Seidel iterations, diagonal preconditioning, domain decomposition multigrid multilevel nodal basis preconditioners hierarchical presented. By using the notions space subspace correction, all these algorithms are classified into two...

In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners discrete systems which arise in numerical approximation symmetric elliptic boundary value problems. These are defined as a sum independent operators on sequence nested subspaces full space. On computer, evaluation these hence preconditioner given function can be computed concurrently. We shall study new technique developing first an abstract setting, next by considering applications...

A new finite element discretization technique based on two (coarse and fine) subspaces is presented for a semilinear elliptic boundary value problem. The solution of nonlinear system the fine space reduced to small (one linear one nonlinear) systems coarse space. It shown, both theoretically numerically, that can be extremely still achieve asymptotically optimal approximation. As result, numerical such equation not significantly more expensive than single linearized equation.

In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations H(curl, )- H(div, )-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson We prove mesh-independent effectivity the by using abstract theory auxiliary space preconditioning. main tools are analogues so-called regular decomposition results in function spaces ) ). Our preconditioner is...

This paper provides an overview of AMG methods for solving large-scale systems equations, such as those from discretizations partial differential equations. is often understood the acronym ‘algebraic multigrid’, but it can also be ‘abstract multigrid’. Indeed, we demonstrate in this how and why algebraic multigrid method better at a more abstract level. In literature, there are many different that have been developed perspectives. try to develop unified framework theory used derive analyse...

In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as activation and continuous piecewise (CPWL) functions, especially CPWL functions from simplicial finite element method (FEM).We first consider special case of FEM.By exploring DNN representation its nodal basis present a ReLU in FEM.We theoretically establish that at least 2 hidden layers are needed to represent any Ω ⊆ R d when ≥ 2. Consequently, for = 2, 3 which...

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an problem on a fine grid reduced to much coarser grid, linear algebraic system resulting still maintains asymptotically optimal accuracy.

In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space "V which are defined in terms products operators with respect to number subspaces.The simplest algorithm sort has an error-reducing operator is product orthogonal projections onto complement subspaces.New normreduction estimates these techniques will be presented abstract setting.Applications given overlapping Schwarz algorithms many subregions finite element approximation...

The purpose of this paper is to give a unified investigation class nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions. under scrutiny fall into major categories: the substructuring--type Neumann--Neumann-type methods. basic framework used analysis parallel subspace correction method or additive Schwarz method, other technical tools include local-global global-local techniques. analyses both two- three-dimensional cases are...

We provide a theory for the analysis of multigrid algorithms symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that form on coarser grids need not be related to finest, i.e., do stay within standard variational setting. In more general setting, give new estimates corresponding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V"> <mml:semantics> <mml:mrow...

We study numerical methods for a coupled Navier–Stokes/Darcy model which describes fluid flow filtrating through porous media. A decoupled and linearized two-grid algorithm is proposed. Numerical analysis experiments are presented to show the efficiency effectiveness of algorithm.

A new technique is proposed to solve NSPD (nonsymmetric or indefinite) problems that are “compact” perturbations of some SPD (symmetric positive definite) problems. In the algorithm, a direct method first used original equation restricted on coarser space (that has considerably smaller dimension), then an for residue solved by using one few iterations given iterative algorithm. It shown any convergent problem, algorithm always converges with essentially same rate if coarse properly chosen....

A new nonlinear Galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations. The scheme two different spaces defined respectively one coarse grid with size H and fine $h \ll H$. Nonlinearity time dependence are both treated the space only a fixed stationary equation needs to be solved at each time. With linear discretizations, it proved that difference between solution standard $H^1 (\Omega )$ norm of order $H^3 $.

This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for linear finite element approximation second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing eigenvalue distribution BPX preconditioner and multigrid V-cycle preconditioner, we prove that only small number eigenvalues may deteriorate respect jump or meshsize, all other are bounded below above nearly uniformly meshsize. As result,...

In Part I of this work [SIAM J. Numer. Anal., 41 (2003), pp. 2294--2312], we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs triangles sharing a common edge form approximate parallelograms. work, consider general unstructured but shape regular meshes. We develop postprocessing gradient recovery scheme the solution uh, inspired in part by smoothing iteration multigrid method. This recovered superconverges to true and becomes...