Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations

Summary. The - spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove that the condition number of the iteration operator is bounded from above by , where C is a positive constant independent of the degree N and the number of subdomains, and is the inf-sup condition of the pair -. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove that the number of iterations of the GMRES method, at each Newton step, is bounded from above by . The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds. Received March 2, 1998 / Revised version received October 12, 1999 / Published online March 20, 2001     

Eigenvalue bounds for the Schur complement with a pressure convection–diffusion preconditioner in incompressible flow computations

If the stationary Navier–Stokes system or an implicit time discretization of the evolutionary Navier–Stokes system is linearized by a Picard iteration and discretized in space by a mixed finite element method, there arises a saddle point system which may be solved by a Krylov subspace method or an Uzawa type approach. For each of these resolution methods, it is necessary to precondition the Schur complement associated to the saddle point problem in question. In the work at hand, we give upper and lower bounds of the eigenvalues of this Schur complement under the assumption that it is preconditioned by a pressure convection–diffusion matrix.     

Projektive Newton-Verfahren und Anwendungen auf nichtlineare Randwertaufgaben

Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods.

     

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.     

Efficient fourth orderP-stable formulae

In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.     

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.     

Diagonally implicit Runge–Kutta methods for 3D shallow water applications

We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized spatial domain is not involved in the stability condition. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences).     

Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods

This paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P₁)-Q₀ elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method.     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface

By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI-DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm.     

A cascadic multigrid algorithm for semilinear elliptic problems

Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear finite element discretizations are solved by Newton's method. Given an approximate solution on the coarsest grid on each finer grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity. Received February 12, 1998 / Revised version received July 22, 1999 / Published online June 8, 2000     

The hierarchical basis multigrid method for convection-diffusion equations

Summary We make a theoretical study of the application of a standard hierarchical basis multigrid iteration to the convection diffusion equation, discretized using an upwind finite element discretizations. We show behavior that in some respects is similar to the symmetric positive definite case, but in other respects is markedly different. In particular, we find the rate of convergence depends significantly on parameters which measure the strength of the upwinding, and the size of the convection term. Numerical calculations illustrating some of these effects are given.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.     

Efficient sixth order methods for nonlinear oscillation problems

A class of two-step (hybrid) methods is considered for solving pure oscillation second order initial value problems. The nonlinear system, which results on applying methods of this type to a nonlinear differential system, may be solved using a modified Newton iteration scheme. From this class the author has derived methods which are fourth order accurate,P-stable, require only two (new) function evaluations per iteration and have a true real perfect square iteration matrix. Now, we propose an extension to sixth order,P-stable methods which require only three (new) function evaluations per iteration and for which the iteration matrix is a true realperfect cube. This implies that at most one real matrix must be factorised at each step. These methods have been implemented in a new variable step, local error controlling code.     

The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

A class of difference ABS-type algorithms for a nonlinear system of equations

In this paper we give a class of algorithms for solving nonlinear algebraic equations using difference approximations of derivatives. The class is a modification of the original ABS class with the advantage of requiring less function evaluations. Special cases include the methods of Brown and Brent and the discretized Newton method, which is formulated in a way requiring fewer function evaluations per iteration.     

Parallel frontal methods for ODE's

This paper concerns parallel frontal predictor-corrector methods. Order and stability of these methods are investigated, when the corrector is solved both by the fixed point iteration method and by the Newton method.This work has been partially supported by the Italian C.N.R. within the Finalized Project Sistemi Informatici e Calcolo Parallelo.     

Additive Schwarz methods for thep-version finite element method

Summary In this paper, we study some additive Schwarz methods (ASM) for thep-version finite element method. We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. We prove a constant bound independent of the degreep and the number of subdomainsN, for the condition number of the ASM iteration operator. This optimal result is obtained first in dimension two. It is then generalized to dimensionn and to a variant of the method on the interface. Numerical experiments confirming these results are reported. As is the case for other additive Schwarz methods, our algorithms are highly parallel and scalable.This work was supported in part by the Applied Math. Sci. Program of the U.S. Department of Energy under contract DE-FG02-88ER25053 and, in part, by the National Science Foundation under Grant NSF-CCR-9204255     

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.     

A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Summary. We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements. Received March 17, 1998 / Revised version received June 7, 1999 / Published online January 27, 2000