Showing papers in "SIAM Journal on Numerical Analysis in 1982"

An Interior Penalty Finite Element Method with Discontinuous Elements

Abstract: A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and $L^2$-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.

Inexact newton methods

Abstract: A classical algorithm for solving the system of nonlinear equations $F(x) = 0$ is Newton’s method \[ x_{k + 1} = x_k + s_k ,\quad {\text{where }}F'(x_k )s_k = - F(x_k ),\quad x_0 {\text{ given}}.\]...

Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures*

Abstract: Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems $(b \gg a)$, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

Abstract: This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier–Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty $. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier–Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.

Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex

Abstract: A simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex.

Newton's method with a model trust region modification

Abstract: A modified Newton method for unconstrained minimization is presented and analyzed. The modification is based upon the model trust region approach. This report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified Newton iteration. Several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis. Very strong convergence results are presented concerning the minimization algorithm. In particular, the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization. With the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives.

Numerical Treatment of Stochastic Differential Equations

Abstract: We define general Runge–Kutta approximations for the solution of stochastic differential equations (sde). These approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift. The explicit form of the correction term is given.Concerning the order of convergence, we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than $O(h^3 )$. This order is attained, e.g., by the stochastic analogue of Heun’s method. In the n-dimensional case, the highest order of convergence is in general only $O(h^2 )$, attained by Euler’s method. The order $O(h^3 )$ can only be reached if $( abla _x \sigma ^r )\sigma ^s = ( abla _x \sigma ^s )\sigma ^r $ for the diffusion matrix.

A Family of Trust Region Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties.

Abstract: This paper has two aims: to exhibit very general conditions under which members of a broad class of unconstrained minimization algorithms are globally convergent in a strong sense, and to propose several new algorithms that use second derivative information and achieve such convergence. In the first part of the paper we present a general trust-region-based algorithm schema that includes an undefined step selection strategy. We give general conditions on this step selection strategy under which limit points of the algorithm will satisfy first and second order necessary conditions for unconstrained minimization. Our algorithm schema is sufficiently broad to include line search algorithms as well. Next, we show that a wide range of step selection strategies satisfy the requirements of our convergence theory. This leads us to propose several new algorithms that use second derivative information and achieve strong global convergence, including an indefinite line search algorithm, several indefinite dogleg algo...

Norm-preserving dilations and their applications to optimal error bounds*

Abstract: The problem is, given A, B, C, to find D such that $\left\| ( {\begin{array}{*{20}c} A & C \\ B & D \\ \end{array} )} \right\| \leqq \mu $; here we deal with Hilbert-space operators, A, B, and C ar...

The Lanczos Biorthogonalization Algorithm and Other Oblique Projection Methods for Solving Large Unsymmetric Systems

Abstract: Many powerful methods for solving systems of equations can be regarded as projection methods. Most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal (or oblique) projection methods, which is particularly attractive for large nonsymmetric systems. The purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results. Some experiments comparing the various algorithms are reported.

A Trace Minimization Algorithm for the Generalized Eigenvalue Problem

Abstract: An algorithm for computing a few of the smallest (or largest) eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem $Ax = \lambda Bx$ is presented. The matrices A and B are assumed to be symmetric, and haphazardly sparse, with B being positive definite. The problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace. The rate of convergence of the method is established, and a technique for improving it is discussed. Numerical experiments and comparisons with other methods are presented.

On the Smoothing Property of the Galerkin Method for Parabolic Equations

Abstract: We analyze the Galerkin approximation of the general second-order parabolic initial-boundary value problem. For the second-order continuous time method with initial data only in $L^2 $ we prove an ...

Existence, Uniqueness and Approximation for Generalized Saddle Point Problems

Abstract: Extensions of the well-known results of Brezzi on saddle point problems are presented. The class of problems is generalized to include the unsymmetric case, and the known stability and approximation results are strengthened, and applied to the generalized problem. An application to the solution of certain exterior interface problems is indicated.

Residual Bounds on Approximate Eigensystems of Nonnormal Matrices

Abstract: For nonnormal matrices the norms of the residuals of approximate eigenvectors are not by themselves sufficient information to bound the error in the approximate eigenvalue. It is sufficient however to give a bound on the distance to the nearest matrix for which the given approximations are exact. This result is extended to cover approximate invariant subspaces and their residuals.The theorems are used to derive a useful set of criteria for terminating the two-sided Lanczos algorithm.The study begins with a list of error bounds for eigenvalues of nonnormal matrices.

On Numerical Boundary Treatment of Hyperbolic Systems for Finite Difference and Finite Element Methods

Abstract: It is well known that the stability of the initial-boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment. In fact, we show that for any boundary treatment one can construct systems for which the boundary treatment on the “natural” variables is not stable. Our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation. This is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary. In a working code this can be accomplished by adding a correction term to the existing boundary algorithm. The analysis and computational results are presented for both finite differences and semi-discrete Galerkin methods.

Multigrid Methods for Variational Problems

Abstract: This paper develops a very simple but powerful theory for multigrid methods which applies directly to variationally posed operator equations.

Properties of Interpolatory Product Integration Rules

Abstract: This paper is concerned with the approximate evaluation of $\int_{ - 1}^1 {k(x)f(x)d_x } $, where k is Lebesgue integrable and f is at least Riemann integrable, and preferably smooth. The integral is approximated by a rule of the form $\sum olimits_{i = 1}^n {w_{ni} (k)f(x_{ni} )} $, where the points $x_{ni} $ are chosen in some prescribed way, and the weights $w_{ni} (k)$ are such that the rule is exact if f is any polynomial of degree $ < n$. For suitable choices of the points and suitable functions k, the rule is shown to converge to the exact integral, and the companion rule $\sum {|w_{ni} (k)|f(x_{ni} )} $ to$\int {|k(x)|f(x)dx} $, for all Riemann integrable functions f. (The companion property ensures, for example, that the weights have the asymptotic positivity property if k is positive.) Error bounds are given that guarantee rapid convergence of the rule if f is smooth. Finally, by establishing a connection with mean convergence of Lagrangian interpolation some negative theorems are proved. In pa...

A Theoretical Framework for Proving Accuracy Results for Deferred Corrections

Abstract: General techniques are described for proving accuracy results for deferred correction solutions to differential equations. These techniques apply also to computational estimates of the local discretization error. The proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter. 1. Introduction. This is a continuation of work begun by Lindberg (26) on error estimation and iterative improvement for the numerical solution of operator equations. Good theoretical and practical results for smooth problems are obtained by Lindberg using what might be regarded as a generalization of Fox's deferred difference correc- tion. (These difference corrections are, in fact, estimates of the local discretization error with the sign reversed.) We have since been able to simplify and strengthen Lindberg's theorems. The result is a theoretical framework for proving accuracy results for deferred corrections, of which local and global error estimation are special cases. Application of this theory to ordinary differential equations yields useful theoretical results, and we hope that the same is possible for partial differential equations. An improved but condensed version of the 1976 report has been written by Lindberg (27). We use the term "deferred correction" in a broad sense to include not only the sort of technique developed by Pereyra (31) but also the "difference correction" of Fox (10) and the "defect correction, version B" described by Stetter (38) in an important paper which gives a general treatment of this topic from a different point of view. Deferred correction has been used very successfully to obtain solutions of improved accuracy for two-point boundary value problems, for example, by Lentini and Pereyra (25) and Daniel and Martin (8). An extension of the deferred correction idea has been used by Stetter (41) to estimate the global error for the state-of-the-art integrator STEP of Shampine and Gordon. Advantages of deferred correction over Richardson extrapolation are given in the opening paragraphs of Stetter (38) and Keller and Pereyra (23). Our error analysis applies also to computational estimates of local errors since this constitutes the first half of a deferred correction step. Local error estimates by themselves are an important component of adaptive mesh refinement and variable time step strategies. A major strength of our approach is that it avoids the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of a meshsize parameter h. In the case of ODEs this means that one can drop the somewhat artificial assumption that there exists a function O(t) such that the nth stepsize hn hO(tn_l)+ o(h), and it enables one to justify more easily error estimates for variable order methods. (However, no examples of this are given in this paper.) It is not the existence of an asymptotic expansion which is crucial but rather the smoothness of the global error. This can be measured by means of "discrete Sobelev" norms defined in terms of divided differences (Lindberg, Skeel, and Van Rosendale (28)). Norms

Efficient Time-Stepping Methods for Miscible Displacement Problems in Porous Media

Abstract: Efficient procedures for time-stepping Galerkin methods for approximating the solution of a coupled system for $c = c(x,t)$ and $p = p(x,t)$, with nonlinear Neumann boundary conditions, of the form \[ \begin{gathered} - abla \cdot [a(x,c)\{ abla p - y(x,c) abla g\} ] \equiv abla \cdot u = f_1 (x,t),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ abla \cdot [b(x,c, abla p) abla c] - u \cdot abla c = \phi (x)\frac{{\partial c}} {{\partial t}} - f_2 (x,t,c),\quad x \in \Omega ,\quad t \in (0,T], \hfill \\ u \cdot v = q_1 (x,t),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ b\frac{{\partial c}} {{\partial v}} = q(x,t,c),\quad x \in \partial \Omega ,\quad t \in (0,T], \hfill \\ c(x,0) = c_0 (x),\quad x \in \Omega , \hfill \\ \end{gathered} \] where $\Omega \subset \mathbb{R}^d $, $2 \leqq d \leqq 3$, are presented and analyzed. This system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a...

On the Linear Independence of Multivariate B-Splines, I. Triangulations of Simploids

Abstract: This paper is concerned with the construction of certain collections of linearly independent multivariate B-splines whose linear span is appropriate for approximation. By constructing linear projectors onto these spline spaces we are able to estimate local and global approximation errors as well as the condition number of the B-spline basis. Our approach is based upon some combinatorial properties of certain triangulations of a class of n-polytopes which we call simploids.

The Classical Collocation Method for Singular Integral Equations

Abstract: First of all a method of collocation, which we call “classical” collocation, is described for the approximate solution of complete singular integral equations with Cauchy kernel taken over the arc $( - 1,1)$. Secondly we demonstrate that, under reasonable conditions, the approximate solutions converge to the solution of the original equation.

Finite-Difference Approximations and Superconvergence for the Discrete-Ordinate Equations in Slab Geometry

Abstract: A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors.

Computation of Critical Boundaries on Equilibrium Manifolds

Abstract: The study of various equilibrium phenomena leads to nonlinear equations (1) $F(y,u) = 0$, where $y \in R^n $ is a vector of behavior or state variables, $u \in R^p $ a vector of $p \geqq 2$ parameters or controls and $F:D \subset R^n \times R^p \to R^n $ a sufficiently differentiable map. The solution set of (1) in $R^n \times R^p $ is often called the equilibrium surface of the problem, and in many applications it is of interest to determine the critical boundary, that is, the set of solutions of (1) where the derivative $D_y F(x)$ is singular. For example, in structural problems these points may represent buckling points. After characterizing the various properties of the problem, we present three different numerical methods which allow for a computational trace of paths in the critical boundary. These methods represent extensions of the earlier-developed locally parametrized continuation method for tracing regular paths on the equilibrium surface. They permit, for instance, a direct computational deter...

Use of Iterative Refinement in the Solution of Sparse Linear Systems

Abstract: It is well known that if Gaussian elimination with iterative refinement (IR) is used in the solution of systems of linear algebraic equations $Ax = b$ whose matrices are dense, then the accuracy of the results will usually be greater than the accuracy obtained by the use of Gaussian elimination without iterative refinement (DS). However, both more storage (about $100\% $, because a copy of matrix A is needed) and more computing time (some extra time is needed to perform the iterative process) must be used with IR. Normally, when the matrix is sparse the accuracy of the solution computed by some sparse matrix technique and IR will still be greater. In this paper it is verified (by many numerical experiments) that the use of sparse matrix techniques with IR may also result in a reduction of both the computing time and the storage requirements (this will never happen when IR is applied for dense matrices). Two parameters, a drop-tolerance $T \geqq 0$ and a stability factor $u > 1$, are introduced in the effo...

An Enthalpty Formulation of the Stefan Problem

Abstract: In this paper the implicit time discretization of $E_t - abla \cdot K abla T = f$, the enthalpy formulation of the Stef an problem, is considered. This generates the algebraic system $E + A\beta (E) = \eta $, where E, $\beta (E)$, $\eta \in {}^ + \mathbb{R}^L $, A is an M-matrix and $\beta (E)$ is the “inverse” of the enthalpy function. The algebraic equation is solved by a modification of the Gauss–Seidel method and convergence is proved. The existence and uniqueness of a weak solution with $T \in W_2^{1,0} (Q_T )$ is established by Rothe’s method. An application to thermal energy storage units which utilize phase change materials is given. Heat transfer via both diffusion and convection of the liquid phase change material is considered.

A Fast Algorithm for Smoothing Data on a Rectangular Grid while Using Spline Functions

Abstract: An efficient computational method is presented for fitting a bivariate spline function to a set of measured data on a rectangular grid. The coefficients in the B-spline representation of this spline are obtained by the solution of a linear system which can be arranged in a matrix form, conformable withthe Kronecker product of two band matrices of small size and bandwidth. The number of knots of the spline and their positions are determined automatically. Instead the algorithm expects a parameter to control the tradeoff between closeness of fit and smoothness of fit.

The Numerical Solution of Laplace’s Equation in Three Dimensions

Abstract: We consider the Dirichlet problem for Laplace’s equation, on a simply-connected three-dimensional region with a smooth boundary. This problem is easily converted to the solution of a Fredholm integral equation of the second kind, based on representing the harmonic solution as a double layer potential function. We solve this integral equation formulation by using Galerkin’s method, with spherical harmonics as the basis functions. This approach leads to small linear systems, and once the Galerkin coefficients for the region have been calculated, the computation time is small for any particular boundary function. The major disadvantage of the method is the calculation of the Galerkin coefficients, each of which is a four-fold integral with a singular integrand. Theoretical and computational details of the method are presented.

On the Numerical Solution of Initial/Boundary-Value Problems in One Space Dimension

Abstract: The numerical solution of initial/boundary-value problems of the form \[ A(u,x,t)u_t + B(u,x,t)u_x = c(u,x,t)\] is considered. Particular emphasis is placed on the solution of problems with large gradients, e.g., shocks and boundary layers. A mesh-selection technique is described that accurately places points in regions where the solution is rapidly changing. This is accomplished by a transformation of the original equations to a new coordinate system. Finite difference solutions for two sample problems are calculated.

On a Galerkin--Lagrange Multiplier Method for the Stationary Navier--Stokes Equations

Abstract: A Galerkin–Lagrange multiplier formulation is used for the numerical solution of the stationary Navier–Stokes equations, in order to avoid the construction of zero-divergence elements. The formulation is based on different approximating spaces for the velocity field and the pressure. Optimal rate of convergence estimates are derived. Moreover, a Galerkin–Newton scheme for the solution of the nonlinear equations is shown to be quadratically locally convergent. Another scheme is shown to be linearly globally convergent.

Spectral and Pseudo-Spectral Approximations of the Navier--Stokes Equations

Abstract: Galerkin and collocation approximations by trigonometric polynomials to the stationary Navier-Stokes equations with periodic boundary conditions are analyzed. Stability results and “optimal” rates of convergence for both velocity and pressure in Sobolev spaces are proven. Essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems [Brezzi, Rappag, Raviart, Numer. Math., 36 (1980), pp. 1–25], [Maday, Quarteroni, RAIRO Numer. Anal. (1980)].