Preconditioners for solving the indefinite linear system for higher order H (curl) discretisations

This paper presents numerical results for a preconditioning technique for the iterative solution of the indefinite linear system of equations obtained when a higher order H (curl) conforming finite element approximation is adopted for the vector wave equation. Numerical results are presented for a model problem that includes a singularity at the origin and dicretisations that include graded meshes and uniform p. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

Application of hp-discontinuous Galerkin finite element methods to the rotating disk electrode problems in electrochemistry

This paper presents the interior penalty discontinuous Galerkin finite element methods (DGFEM) for solving the rotating disk electrode problems in electrochemistry. We present results for the simple E reaction mechanism (convection-diffusion equations), the EC’ reaction mechanism (reaction-convection-diffusion equation) and the ECE and EC₂E reaction mechanisms (linear and nonlinear systems of reaction-convection-diffusion equations, respectively). All problems will be in one dimension.     

A finite element — Capacitance method for elliptic problems on regions partitioned into subregions

Summary A method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem. This method reduces the original problem to several subproblems (of the same form) considered on subregions, and an auxiliary problem. Very efficient iterative methods with the preconditioning operator and using FFT are developed for the auxiliary problem.     

Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws

In this paper, we describe a novel formulation of a preconditioned BiCGSTAB algorithm for the solution of ill-conditioned linear systems Ax=b. The developed extension enables the control of the residual r _m =b–Ax _m of the approximate solution x _m independent of the specific left, right or two-sided preconditioning technique considered. Thereby, the presented modification does not require any additional computational effort and can be introduced directly into existing computer codes. Furthermore, the proceeding is not restricted to the BiCGSTAB method, hence the strategy can serve as a guideline to extend similar Krylov sub-space methods in the same manner. Based on the presented algorithm, we study the behavior of different preconditioning techniques. We introduce a new physically motivated approach within an implicit finite volume scheme for the system of the Euler equations of gas dynamics which is a typical representative of hyperbolic conservation laws. Thereupon a great variety of realistic flow problems are considered in order to give reliable statements concerning the efficiency and performance of modern preconditioning techniques.     

Mixed finite element methods for incompressible flow: Stationary Stokes equations

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress‐velocity formulation. The pseudostress is approximated by the Raviart‐Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi‐optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Numerical simulations of glacial rebound using preconditioned iterative solution methods

This paper discusses finite element discretization and preconditioning strategies for the iterative solution of nonsymmetric indefinite linear algebraic systems of equations arising in modelling of glacial rebound processes. Some numerical experiments for the purely elastic model setting are provided. Comparisons of the performance of the iterative solution method with a direct solution method are included as well.     

Two-Grid hp-Version DGFEMs for Strongly Monotone Second-Order Quasilinear Elliptic PDEs

In this article we develop the a priori error analysis of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of strongly monotone second-order quasilinear partial differential equations. In this setting, the fully nonlinear problem is first approximated on a coarse finite element space V(𝒯_H, P ). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearize the underlying discretization on the finer space V(𝒯_h, p ); thereby, only a linear system of equations is solved on the richer space V(𝒯_h, p ). Numerical experiments confirming the theoretical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Inverse problem for N × N hyperbolic systems on the plane and the N-wave interactions

We study regiorously the solvability of the direct and inverse problems associated with Ψ_x–JΨ_y = QΨ,(x,y) ∈ ?², where (i) Ψ is an N × N-matrix-valued function on ?² (N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J₁ > J₂ > …? > J_N and (iii) Q is off-diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N-wave interactions and show that if a certain norm of Q(x, y, 0) is smallor if Q(x, y, 0) is skew Hermitian the N-wave interations equation has a unique global solution.     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h _n : 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {V _Ψ (〈c, d〉; f ? h _n )} _n=1 ^∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ? with finite Φ-variation and the functions h _n . In Theorem 2, the same is done for a sequence of functions h _n : ? → ?, n = 1, 2, ..., and the sequence of Ψ-variations {V _Ψ(〈a, b〉; h _n ? f)} _n=1 ^∞ .     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Parallel solution of elasticity problems using overlapping aggregations

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper.     

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

A capacitance matrix method for Dirichlet problem on polygon region

Summary An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN ²log₂ N (N=1/h) where the cost of solving the capacitance matrix equations isNlog₂ N on regular grids andN ^3/2log₂ N on irregular ones.     

Transient solutions of a software model with imperfect debugging and generation of errors by two servers

In this paper, we obtain the transient solution of probabilities of error in the software, mean number of faults and the expected number of failures remaining at time t, under the assumption that the number of faults is finite, the failure rate is proportional to the number of faults present in the software at any time, debugging is imperfect and error generation will never lead the software to have infinite errors. Moreover, the software is tested by two servers with the first M errors being debugged by first server and the remaining errors (M +1 ≤n ≤N) by the second server. Also, when a failure occurs, instantaneously repair starts with the following probabilities.

• 1.(a) The fault content is reduced by one by the first (second) server with probability μ₁(μ₂),μ₂ ≥μ₁
• 2.(b) The fault content remains unchanged with probability Ψ.
• 3.(c) The fault content is increased by one by the first (second) server with probability λ₁(λ₂), λ₁ ≥λ₂ where μ₁ + Ψ + λ₁ = 1, μ₂ + Ψ + λ₂ = 1, μ₁ ⪢ Ψ ⪢ λ₁, μ₂ ⪢ Ψ ⪢ λ₂. Finally, a numerical example is presented for the transient probabilities for the number of errors in the software, mean number of faults and the expected number of failures remaining in the software.