Showing papers in "SIAM Journal on Numerical Analysis in 1995"

Implicit-explicit methods for time-dependent partial differential equations

Abstract: Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes, and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods.For the prototype linear advection-diffusion equation, a stability analysis for first-, second-, third-, and fourth-order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay...

Adaptive finite element methods for parabolic problems IV: nonlinear problems

Abstract: We extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems. We prove a posteriori error estimates, design corresponding adaptive algorithms, and present some numerical results.

Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞

Abstract: Optimal error estimates are derived for a complete discretization of linear parabolic problems using space–time finite elements. The discretization is done first in time using the discontinuous Galerkin method and then in space using the standard Galerkin method. The underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step. The error bounds show, in particular, that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same (given) level, i.e., without error accumulation effects. The derivation of the estimates is based on the orthogonality of the Galerkin procedure and the use of strong stability estimates. The particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration.

Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation

Abstract: In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some ...

A characteristics-mixed finite element method for advection-dominated transport problems

Abstract: We define a new finite element method, called the characteristics-mixed method, for approximating the solution to an advection-dominated transport problem. The method is based on a space-time variational form of the advection-diffusion equation. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the advective (i.e., hyperbolic) part of the equation. Thus the scheme uses a characteristic approximation to handle advection in time. This is combined with a low-order mixed finite element spatial approximation of the equation. Boundary conditions are incorporated in a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. A postprocessing step is included in the scheme in which the approximation to the scalar unknown is improved by utilizing the approximate vector flux. This has the effect of improving the rate of convergence of ...

Projection method I: convergence and numerical boundary layers

Abstract: This is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of these papers is to provide a thorough understanding of the ...

A posteriori error bounds and global error control for approximation of ordinary differential equations

Abstract: The author analyzes a finite element method for the integration of initial value problems in ordinary differential equations. General and contractive problems are treated, and quasi-optimal a priori and a posteriori error bounds obtained in each case. In particular, good results are obtained for a class of stiff dissipative problems. These results are used to construct a rigorous and robust theory of global error control. The author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error. In conclusion, the properties of the error control are exhibited in a series of numerical experiments.

A Simple Proof of Convergence for an Approximation Scheme for Computing Motions by Mean Curvature

Abstract: We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions methods.

Error estimates on a new nonlinear Galerkin method based on two-grid finite elements

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

Finite element approximation of time harmonic waves in periodic structures

Abstract: Consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material. It is shown that the scattering (diffraction) problem may be modeled by a Helmholtz equation with transparent boundary conditions. The diffraction problem may be solved by a finite element method. In this paper, optimal error estimates for the finite element method are established. The error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place.

On error estimates of the penalty method for unsteady Navier-Stokes equations

Abstract: The penalty method has been widely used for numerical computations of the unsteady Navier–Stokes equations. However, the best error estimates available to the author’s knowledge were not optimal an...

A general class of two-step Runge-Kutta methods for ordinary differential equations

Abstract: A general class of two-step Runge–Kutta methods that depend on stage values at two consecutive steps is studied. These methods are special cases of general linear methods introduced by Butcher and are quite efficient with respect to the number of function evaluations required for a given order. General order conditions are derived using the approach proposed recently by Albrecht, and examples of methods are given up to the order 5. These methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments.

Relative perturbation techniques for singular value problems

Abstract: A technique is presented for deriving bounds on the relative change in the singular values of a real matrix (or the eigenvalues of a real symmetric matrix) due to a perturbation, as well as bounds on the angles between the unperturbed and perturbed singular vectors (or eigenvectors). The class of perturbations considered consists of all $\delta B$ for which $B + \delta B = D_L BD_R $ for some nonsingular matrices $D_L $ and $D_R $. This class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off-diagonal block in a block triangular matrix. Many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations. Also some new relative perturbation and deflation results for the singular values and vectors of biacyclic, triangular, and shifted triangular matrices are presented.

Finite element vibration analysis of fluid-solid systems without spurious modes

Abstract: This paper deals with the finite element approximation of the vibration modes of a problem with fluid–structure interaction Displacement variables are used for both the fluid and the solid To avoid the typical spurious modes of this formulation we introduce a nonconforming discretization Error estimates for the approximation of eigenvalues and eigenvectors are given

Mixed finite element methods for nonlinear second-order elliptic problems

Abstract: Mixed finite element methods are developed to approximate the solution of the Dirichlet problem for the most general quasi-linear second-order elliptic operator in divergence form. Existence and uniqueness of the approximation are proved, and optimal error estimates in $L^2 $ are demonstrated for both the scalar and vector functions approximated by the method. Error estimates are also derived in $L^q $, $2 \leq q \leq + \infty $. Newton’s method is presented and analyzed to solve the nonlinear algebraic equations.

Numerics and hydrodynamic stability: toward error control in computational fluid dynamics

Abstract: We critically review the available error analysis in computational fluid dynamics (CFD) and come to the conclusion that the existing error estimates are meaningless in most cases of interest. We propose a new approach to error analysis in CFD aiming at reliable and efficient adaptive quantitative error control. This is based on a precise analysis of hydrodynamic stability coupled with Galerkin orthogonality. We prove a priori- and a posteriori-type error estimates in a model case for pipe flow, formulate corresponding adaptive algorithms, and discuss the potential of this approach for adaptive error control in CFD.

A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets

Abstract: This paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach. An axiomatic approach of image processing and the mean curvature partial differential equation (PDE) are briefly presented, then the properties of the proposed scheme are studied. In particular, consistency and convergence are proved. The applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented.

Adaptive finite element methods for parabolic problems V: long-time integration

Abstract: We continue our previous work on adaptive finite element methods for parabolic problems, now with particular emphasis on long-time integration for semidefinite problems.

Stability and convergence of a class of enhanced strain methods

Abstract: A stability and convergence analysis is presented of a recently proposed variational formulation and finite element method for elasticity, which incorporates an enhanced strain field. The analysis is carried out for problems posed on polygonal domains in $R^n $, the finite element meshes of which are generated by affine maps from a master element. The formulation incorporates as a special case the classical method of incompatible modes. The problem initially has three variables, viz, displacement, stress, and enhanced strain, but the stress is later eliminated by imposing a condition of orthogonality with respect to the enhanced strains. Two other conditions on the choice of finite element spaces ensure that the approximations are stable and convergent. Some features of nearly incompressible and incompressible problems are also investigated. For these cases it is possible to argue that locking will not occur, and that the only spurious pressures present are the so-called checkerboard modes. It is shown th...

Stepwise stability for the heat equation with a nonlocal constraint

Abstract: The authors prove the stepwise stability for a finite difference scheme for the heat equation with an integral constraint. The resulting matrix is nonsymmetric and does not have the usual band structure. The proof is based on the method of matrix analysis. The eigenvalues of several matrices are found explicitly or their location described precisely. This method relies upon the relationship of the characteristic polynomials of these matrices with orthogonal polynomials.

The numerical solution of delay-differential-algebraic equations of retarded and neutral type

Abstract: In this paper we consider the numerical solution of initial-value delay-differential-algebraic equations (DDAEs) of retarded and neutral types, with a structure corresponding to that of Hessenberg DAEs. We give conditions under which the DDAE is well conditioned and show how the DDAE is related to an underlying retarded or neutral delay-ordinary differential equation (DODE). We present convergence results for linear multistep and Runge–Kutta methods applied to DDAEs of index 1 and 2 and show how higher-index Hessenberg DDAEs can be formulated in as stable a way as index-2 Hessenberg DDAEs. We also comment on some practical aspects of the numerical solution of these problems.

Convergence of the finite volume method for multidimensional conservation laws

Abstract: We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations (“flat elements” are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169–210 and J. Numer. Anal., 30 (1993), pp. 675–700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna’s uniqueness result in the class of entropy measure-valued solutions.

The potential for parallelism in Runge-Kutta methods. Part 1: RK formulas in standard form

Abstract: The authors examine the potential for parallelism in Runge–Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of an RK formula in terms of the minimal polynomial associated with its coefficient matrix. The positive results are largely examples of prototypical formulas that offer a potential for effective “coarse-grain” parallelism on machines with a few processors.

C 1 -surface splines

Abstract: The construction of quadratic $C^1 $ surfaces from B-spline control points is generalized to a wider class of control meshes capable of outlining arbitrary free-form surfaces in space. Irregular meshes with nonquadrilateral cells and more or less than four cells meeting at a point are allowed so that arbitrary free-form surfaces with or without boundary can be modeled in the same conceptual frame work as tensor-product B-splines. That is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local, evaluates by averaging, and obeys the convex hull property. For a regular region of the input mesh, the representation reduces to the standard quadratic spline. In general, a surface spline is represented by Bernstein–Bezier patches of degree two and three with derivatives matching across boundaries after local reparametrization. According to the user’s choice, these patches can be polynomial or rational, and three-sided, four-sided, or a combination thereof.

Accurate upwind methods for the Euler equations

Abstract: A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate and can be considered as extensions of Godunov’s scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer’s scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the “smooth” part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope-steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new...

On the finite element method for mixed variational inequalities arising in elastoplasticity

Abstract: We analyze the finite-element method for a class of mixed variational inequalities of the second kind, which arises in elastoplastic problems. An abstract variational inequality, of which the elast...

Numerical stability and efficiency of penalty algorithms

Abstract: Penalty algorithms have been somewhat forgotten due to numerical instabilities once believed to be inherent to those methods. One usually has to solve a sequence of such problems, and when the penalty factor decreases too fast, the subproblems may become intractable. Moreover, as the penalty factor decreases, the unconstrained subproblem becomes ill conditioned, and thus difficult to solve. Also, in several intermediate computations, numerical instability may show up. The author proposes remedies to such problems and presents a wide class of numerically stable penalty algorithms. The work is done in the more general context of variational inequality problems, which encompasses optimization problems. The author’s results yield a family of globally convergent, two-step superlinearly convergent, numerically stable algorithms for variational inequality problems. Finally, issues in the numerically stable implementation of intermediate computations within those algorithms are discussed.

On optimal solution of interval linear equations

Abstract: For interval linear algebraic systems Ax =b, we consider the problem of component- wise evaluation of the set B33(A, b) ={A-1b I A E A, b E b } formed by all solutions of Ax = b when A and b vary independently in A and b, respectively. An iterative PSS algorithm is introduced that computes optimal (exact) componentwise estimates of S33 and its convergence is proved under fairly general conditions on the interval system. We introduce the concept of a sequentially guaran- teeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time.

Computation of invariant tori by the method of characteristics

Abstract: In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.

A product-decomposition bound for Bezout numbers

Abstract: Most polynomial systems that arise in practice are not completely general but have special structures. A common form is that each equation must be a sum of products, where each factor has an identifiable generic type. A theorem is proven for such systems which offers a method for obtaining a tighter upper bound on the number of nonsingular solutions than is generally available. At the same time, this theorem provides an approach for solving such systems via polynomial continuation, which results in less computational work. To illustrate the practical usefulness of these ideas, we show that a significant design-of-mechanisms problem can be solved with an order of magnitude less work than the published solution.