Papers by Ludmil Zikatanov
Abstract. This paper contains construction and analysis a finite element approximation for convec... more Abstract. This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in Rd, d≥ 2. This construction is quite close to the scheme of Xu and Zikatanov [22] where a diagonal coefficient matrix has been considered. The scheme is of the class of exponentially fitted methods that does not use upwind or checking the flow direction.
We consider the discrete system resulting from mixed finite element approximation of a second-ord... more We consider the discrete system resulting from mixed finite element approximation of a second-order elliptic boundary value problem with Crouzeix–Raviart non-conforming elements for the vector valued unknown function and piece-wise constants for the scalar valued unknown function. Since the mass matrix corresponding to the vector valued variables is diagonal, these unknowns can be eliminated exactly.
Abstract. Let u and uVn be the solution and, respectively, the finite element solution of the Poi... more Abstract. Let u and uVn be the solution and, respectively, the finite element solution of the Poisson's equation∆ u= f with zero boundary conditions. We construct for any m∈ N and any polygon P a sequence of finite dimensional subspaces Vn such that u− uVn H1≤ C dim (Vn)− m/2 fHm− 1, where f∈ Hm− 1 (P) is arbitrary and C is a constant that depends only on P (we do not assume u∈ Hm+ 1 (P)).
Abstract The convergence analysis on the general iterative methods for the symmetric and positive... more Abstract The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sufficient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided.
This is a further development of [9] regarding multilevel preconditioning for symmetric interior ... more This is a further development of [9] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either continuous or piecewise constant functions.
Abstract An overview of multilevel methods on unstructured grids for elliptic problems will be gi... more Abstract An overview of multilevel methods on unstructured grids for elliptic problems will be given. The advantages which make such grids suitable for practical implementations are exible approximation of the boundaries of complicated physical domains and the ability to adapt the mesh to resolve ne-scaled structures in the solution.
Abstract. In this paper we generalize the analysis of classical multigrid and two-level overlappi... more Abstract. In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition.
This paper is on the construction of energy-minimizing coarse spaces that obey certain functional... more This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems.
Abstract. Using phase field methods, we introduce a penalty formulation for restricting the suppo... more Abstract. Using phase field methods, we introduce a penalty formulation for restricting the support of solutions of the hydrodynamic Poisson-Nernst-Plank equations to evolving subregions of the domain. The formulation is derived through variational principles from a free energy involving the phase field and electrostatic energy. We validate the model by energetic arguments and several dynamic, finite element simulations of the (linear) Navier-Stokes, Poisson-Nernst-Plank and Allen-Cahn system.
This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-p... more This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law.
Abstract: This paper presents estimates of the convergence rate and complexity of an algebraic mu... more Abstract: This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level convergence rate where the coarse level graph is obtained by matching.
Abstract: This work is on the numerical approximation of incoming solutions to Maxwell's equation... more Abstract: This work is on the numerical approximation of incoming solutions to Maxwell's equations with dissipative boundary conditions whose energy decays exponentially with time. Such solutions are called asymptotically disappearing (ADS) and they play an importarnt role in inverse back-scatering problems. The existence of ADS is a difficult mathematical problem.
SUMMARY This paper is on the convergence analysis for two-grid and multigrid methods for linear s... more SUMMARY This paper is on the convergence analysis for two-grid and multigrid methods for linear systems arising from conforming linear finite element discretization of the second-order elliptic equations with anisotropic diffusion. The multigrid algorithm with a line smoother is known to behave well when the discretization grid is aligned with the anisotropic direction; however, this is not the case with a nonaligned grid. The analysis in this paper is mainly focused on two-level algorithms.
Abstract We develop a simple algorithmic framework to solve large-scale linear systems. At its co... more Abstract We develop a simple algorithmic framework to solve large-scale linear systems. At its core, the framework relies on two components:(1) a norm-convergent iterative method and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods.
This paper presents an adaptive algebraic multigrid setup algorithm for positive definite linear ... more This paper presents an adaptive algebraic multigrid setup algorithm for positive definite linear systems arising from discretizations of elliptic partial differential equations. The proposed method uses compatible relaxation to select the set of coarse variables. The nonzero supports for the coarse-space basis are determined by approximation of the so-called two-level “ideal” interpolation operator. Then, an energy minimizing coarse basis is formed using an approach aimed to minimize the trace of the coarse-level operator. The variational multigrid solver resulting from the presented setup procedure is shown to be effective, without the need for parameter tuning, for some problems where current algorithms exhibit degraded performance.
A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By... more A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of space. Our algorithm seems to be able to solve, for example, the Poisson equation discretized on any shape-regular finite element grids with nearly optimal complexity. In this paper, we discuss a multilevel method applied to problems on ...
The method of alternating projections and the method of subspace corrections are general iterativ... more The method of alternating projections and the method of subspace corrections are general iterative methods that have a variety of applications. The method of alternating projections, first proposed by von Neumann (1933) (see ), is an algorithm for finding the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. The method of subspace corrections, an abstraction of general linear iterative methods such as multigrid and domain decomposition methods, is an algorithm for finding the solution of a linear system of equations. In this paper, we shall study these two methods in a Hilbert space setting and in particular present a new identity for the product of nonexpansive operators that gives a sharpest possible estimate of the convergence rate of these methods.
Mathematics of Computation, Jan 1, 2008
The results in this paper are on the convergence rate estimate for the method of successive subsp... more The results in this paper are on the convergence rate estimate for the method of successive subspace corrections applied to semidefinite (singular) problems. We introduce new conditions in the general Hilbert space framework, under which we obtain the convergence rate identity for the method of subspace corrections in terms the subspace solvers. These conditions are in some sense necessary and sufficient for the energy norm convergence of some of the classical iterative methods for the semidefinite problems (see ). To illustrate the results in more general case, we prove a uniform convergence result for the standard multigrid method, when applied to the Laplace equation with pure Neumann boundary conditions.
We announce a well-posedness result for the Laplace equation in weighted Sobolev spaces on polyhe... more We announce a well-posedness result for the Laplace equation in weighted Sobolev spaces on polyhedral domains in R n with Dirichlet boundary conditions. The weight is the distance to the set of singular boundary points. We give a detailed sketch of the proof in three dimensions.
We discuss estimates for the rate of convergence of the method of successive subspace corrections... more We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections. AMS subject classifications: 65F10, 65J05, 65N12, 65N55 Key words: Method of subspace corrections, preconditioning convergence rate of linear iterative method.
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Papers by Ludmil Zikatanov