Showing papers in "SIAM Journal on Numerical Analysis in 1981"
TL;DR: This paper addresses the basic problems of the p-version for the parabolic equation with both variables, x and t discreted via p-versions, and concentrates on the case when in the time variables only one interval is used.
Abstract: In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached.In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inve...
632 citations
TL;DR: The generalized singular value decomposition of any two matrices having the same number of columns has been studied in this paper, where a form for, and a constructive derivation of, the generalized singular values decomposition for any two vectors having columns is given.
Abstract: We suggest a form for, and give a constructive derivation of, the generalized singular value decomposition of any two matrices having the same number of columns We outline its desirable characteristics and compare it to an earlier suggestion by Van Loan [SIAM J Numer Anal, 13 (1976), pp 76–83] The present form largely follows from the work of Van Loan, but is slightly more general and computationally more amenable than that in the paper cited We also prove a useful extension of a theorem of Stewart [SIAM Rev 19 (1977), pp 634–662] on unitary decompositions of submatrices of a unitary matrix
553 citations
TL;DR: This paper presents the theoretical and computational details of the scheme, along with computational trial runs for Burgers’ equation showing that the nodes do concentrate sharply and move with the shocks as desired.
Abstract: We present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or “near-shocks”. These methods allow many nodes automatically to concentrate in the critical regions and move with them. For clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes, with Burgers’ equation as our test equation; but the generalization to much more general spaces and equations (including even certain previous “moving vorticity blobs” of the first author and S. Doss for the Navier–Stokes equations) becomes clear. In this paper we present the theoretical and computational details of our scheme, along with computational trial runs for Burgers’ equation showing that the nodes do concentrate sharply and move with the shocks as desired. The conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a Newton’s method for our implicit stiff ODE solver and were thus limited to ver...
472 citations
TL;DR: In this article, a method for estimating solutions of Fredholm integral equations of the first kind, given noisy data, is presented, where regularization is effected by a smoothing term which is the $L^2 $-norm of the estimate.
Abstract: A method is presented for estimating solutions of Fredholm integral equations of the first kind, given noisy data. Regularization is effected by a smoothing term which is the $L^2 $-norm of the estimate. We propose a scheme by which an approximately optimal amount of smoothing may be computed, based only on the data and the assumed known noise variances. Numerical examples are given for estimating inverse Laplace transforms.
396 citations
TL;DR: In this article, the convergence rate for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations was investigated, and it was shown that if an energy estimate holds, then the extra boundary conditions can be of one order lower accuracy without destroying the convergence expected from the approximation at inner points.
Abstract: The convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations. It is shown that if an energy estimate holds, then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points. If the maximal order of the derivatives occurring in the boundary conditions is low enough, then even lower accuracy can be permitted for the extra boundary conditions.
154 citations
TL;DR: In this paper, a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means is presented, which can be viewed as generalizations of those given by Broyden-Dennis-More [J. Inst. Math. Appl. Comp., 28 (1974), pp. 549-560].
Abstract: The purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means. The theorems and proofs given here can be viewed as generalizations of those given by Broyden–Dennis–More [J. Inst. Math. Appl. 12 (1973), pp. 223–246] and by Dennis–More [Math. Comp., 28 (1974), pp. 549–560]. The analysis is done in the orthogonal projection setting of Dennis–Schnabel [SIAM Rev., 21(1980), pp. 443–459] and many readers might feel that it is easier to understand. The theorems here readily imply local and q-superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of Marwil and Toint and the nonlinear least-squares method of Dennis–Gay–Welsch.
125 citations
TL;DR: In this article, a complete analysis of self-similar solutions for the converging shock and collapsing cavity problems, in spherical and cylindrical geometry, for a perfect gas with arbitrary adiabatic exponent, is presented.
Abstract: A complete analysis is attempted of the self-similar solutions for the converging shock and collapsing cavity problems, in spherical and cylindrical geometry, for a perfect gas with arbitrary adiabatic exponent $\gamma > 1$. Emphasis is given to the rich variety of previously neglected nonanalytic soulutions, and to a full exploration of the relevant parameter space. Distinctions are made between what must be determined numerically and what can be derived algebraically. New solutions are described which contain additional converging shocks, arriving at the origin concurrently with the initial shock or free surface. Some of these new solutions are entirely analytic, except at the shocks themselves, and some are not; in some cases, only one secondary shock is possible, in other cases an arbitrary number. The physical significance of previously rejected partial solutions is discussed. The stability of solutions is discussed in a narrow (one-dimensional) sense. Finally, a study is urged of the asymptotic appr...
122 citations
TL;DR: The paper develops a theory of a posteriors error estimates under the L_p-energy norm for finite element solutions of one-dimensional problems based on a general concept of error indicators and error estimators.
Abstract: The paper develops a theory of a posteriors error estimates under the $L_p $-energy norm for $2 \leqq p \leqq \infty $. The theory is based on a general concept of error indicators and error estimators. Several specific examples of these quantities are introduced and analyzed in detail. The results provide a variety of easily computable error estimates under all these important norms for finite element solutions of one-dimensional problems.
115 citations
TL;DR: In this article, a useful concept of convergence for nonlinear stiff ODEs was developed, which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem.
Abstract: A useful concept of convergence for nonlinear stiff ODE’s will be developed, which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem. This concept will be discussed for three simple methods for stiff systems, the implicit Euler scheme, the implicit midpoint rule and the implicit trapezoidal rule.
110 citations
TL;DR: In this article, it was shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors, which is used for the design of various stepalgorithms for continuation processes; two of them are based on a Newton-corrector while the third o...
Abstract: The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third o...
98 citations
TL;DR: A class of second derivative extended backward differentiation formulas suitable for the approximate numerical integration of stiff systems of first-order ordinary differential equations is examined in this paper. But their approach is restricted to the case of linear systems.
Abstract: A class of second derivative extended backward differentiation formulas suitable for the approximate numerical integration of stiff systems of first-order ordinary differential equations is examine
TL;DR: This paper constructs Runge–Kutta methods of orders 5 and 6 which possess order, stability and efficiency of implementation and classifies all algebraically stable methods of an arbitrary order.
Abstract: There are three interesting properties of methods for (stiff) ordinary differential equations: order, stability and efficiency of implementation. This paper constructs Runge–Kutta methods of orders 5 and 6 which possess these properties to a high extent. We further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods.
TL;DR: In this article, the two-point boundary value problem for a stiff system of ordinary differential equations is considered and an adaptive method to solve these problems even when turning points are present is discussed.
Abstract: Consider the two-point boundary value problem for a stiff system of ordinary differential equations. An adaptive method to solve these problems even when turning points are present is discussed.
TL;DR: For a scalar conservation law, the first order upwind difference scheme is a special case of the Godonov's method as discussed by the authors, which is equivalent to solving a sequence of Riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level.
Abstract: For a scalar conservation law $u_t\ = {f(u)}_x\ with f"$ of constant sign, the first order upwind difference scheme is a special case of Godonov''s method. The method is equivalent to solving a sequence of Riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level. The difference scheme is stable (and the solutions to the associated sequence of Riemann problems do not interact) provided the Courant number $
u$ is less than 1. By allowing and explicitly handling such interactions, it is possible to obtain a generalized method which is stable for $
u$ much larger than 1. In many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods. In particular, shocks can be correctly computed with virtually no smearing. The generalized method is rather unorthodox and still has some problems associated with it. Nonetheless, preliminary results are quite encouraging.
TL;DR: Two multilevel iterative procedures for solving linear systems arising from certain finite element discretizations of nonself-adjoint and indefinite elliptic partial differential equations are described and analyzed and convergence and optimal work estimate are proved.
Abstract: We describe and analyze two multilevel iterative procedures for solving linear systems arising from certain finite element discretizations of nonself-adjoint and indefinite elliptic partial differe...
TL;DR: By perturbing a linear program to a quadratic program, it is possible to solve the latter in its dual variable space by iterative techniques such as successive over-relaxation (SOR) methods.
Abstract: By perturbing a linear program to a quadratic program, it is possible to solve the latter in its dual variable space by iterative techniques such as successive over-relaxation (SOR) methods. This provides a solution to the original linear program.
TL;DR: In this article, a simple difference scheme was proposed to approximate the transport term of the boundary value problem with the same boundary conditions and an arbitrary initial guess and proved convergence to a unique steady state solution as the numerical artificial time approaches infinity.
Abstract: We consider the boundary value problem $\varepsilon y'' - a(y)y' - b(x,y) = F(x), - 1 \leqq x \leqq 1$, for $y( - 1)$ and $y(1)$ given and $b(x,0) \equiv 0$, $b_y (x,y) \geqq 0$. We construct a simple difference scheme approximating $\varepsilon u_{xx} - a(u)u_x - b(x,u) - F(x) = u_t $ for $t \geqq 0, - 1 \leqq x \leqq 1$ with the same boundary conditions and an arbitrary initial guess. As the numerical artificial time approaches infinity, we prove convergence to a unique steady state solution. As the mesh size $\Delta x$ and the parameter $\varepsilon $ approach zero in any order, we prove convergence to the solution of the corresponding O.D.E. We use one-sided differences to approximate the transport term. The method applies equally well to multidimensional analogues. We present numerical results verifying the theory.
TL;DR: In this paper, the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated, and it is shown that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers'' width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, was used in between.
Abstract: The application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated. Collocation at Gauss, Radau and Lobatto points is considered, for both initial and boundary value problems for first order systems with constant coefficients. Particular attention is paid to symmetric schemes for boundary value problems; these problems may have boundary layers at both interval ends. .br Our analysis shows that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers'' width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, is used in between. Ways to construct appropriate layer meshes are proposed. Of all methods considered, the Lobatto schemes appear to be the most promising class of methods, as they essentially retain their usual superconvergence power for the smooth, reduced solution, whereas Gauss-Legendre schemes do not. .br We also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem. For a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number, which indicates that these systems can be solved safely by Gaussian elimination with scaled partial pivoting.
TL;DR: Decker and Kelley as discussed by the authors proved the convergence of Newton's method when the dimension of the null space of the Jacobian matrix is $>1$ and improved on previous results of Decker and Kelley (SIAM J Numer Anal, 17 (1980), pp 66'70) and Reddien (Comput Math Appl, 5 (1980, pp 79'86)
Abstract: A theorem is proved concerning the convergence of Newton’s method when the dimension of the null space of the Jacobian matrix is $>1$ It improves on previous results of Decker and Kelley (SIAM J Numer Anal, 17 (1980), pp 66’70) and Reddien (Comput Math Appl, 5 (1980), pp 79’86)
TL;DR: A vector basis is defined for vector functions that are given on the sphere as discussed by the authors, and it is shown that the basis is orthogonal, that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in Cartesian coordinates.
Abstract: A vector basis is defined for vector functions that are given on the sphere. It is shown that the basis is orthogonal, that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in Cartesian coordinates. The method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics. A vector function that is smooth in Cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates. This is a result of the discontinuous spherical coordinate system rather than the vector function itself. This in turn induces singularities in certain surface derivatives on the sphere, even though the function is bounded and differentiable in Cartesian coordinates. Many of the individual terms in a partial differential equation on the sphere are unbounded at the poles; however, the cancellation between these terms is sufficient to yield a bounded resu...
TL;DR: In this article, the authors generalize the preconditioning techniques introduced by Evans [2] for the numerical solution of the linear system and define two extrapolated versions of the Gauss-Siedel method as well as an extrapolated version of the successive overrelaxation method.
Abstract: This paper generalizes the preconditioning techniques introduced by Evans [2] for the numerical solution of the linear system $Au = b$ and defines two extrapolated versions of the Gauss–Siedel method as well as an extrapolated version of the successive overrelaxation method. Finally, it establishes some convergence theorems for the considered iterative schemes when A has particular properties such as being consistently ordered, positive definite having weak diagonal dominance, etc...
TL;DR: In this article, a linear autonomous functional-differential equations of neutral type are approximated by sequences of ordinary differential equations of increasing dimensions, using cubic splines and cubic Hermite splines.
Abstract: Based on an abstract approximation theorem for ${\text{C}}_0 $-semigroups (Trotter–Kato theorem) we present an algorithm where linear autonomous functional-differential equations of neutral type are approximated by sequences of ordinary differential equations of increasing dimensions. Numerical examples using cubic splines and cubic Hermite splines illustrate the theoretical results.
TL;DR: In this article, the authors study the application of approximation methods, in the general form $F_h (x_h ) = 0, to approximate nonisolated solutions which are geometrically isolated.
Abstract: A solution $x = x^0 $ of $F(x)=0 $ is said to be “isolated” if the Frechet derivative $F'(x^0 )$ is nonsingular. It is said to be “geometrically isolated” if no other solution is in $\| {x - x^0 } \| \leqq \rho $ for some $\rho > 0$. Isolated solutions are always geometrically isolated. Sufficient conditions are obtained to insure that a nonisolated solution is also geometrically isolated. We then study the application of approximation methods, in the general form $F_h (x_h ) = 0$, to approximate nonisolated solutions which are geometrically isolated. Under strong consistency conditions the results are somewhat negative—the approximations may have an even number (including zero) or an odd number of roots near $x^0$, depending upon the “multiplicity” cf $ x^0 $ as a root. If the accuracy is $O(h^p )$ and the multiplicity is N, then the approximations have error $O(h^{p /N})$. The relation of these results to limit and bifurcation points is discussed briefly.
TL;DR: In this article, a combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems and its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension, electrochemical machining and Hele-Shaw flow in two dimensions, and a Stefan and ablation problem in three dimensions.
Abstract: A combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems. Its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension, electrochemical machining and Hele–Shaw flow in two dimensions, and a Stefan and ablation problem in three dimensions.
TL;DR: Galerkin alternating-direction procedures are considered for the nonlinear parabolic equation and an approximation to the Jacobian of the isoparametric map which is based on patches of finite elements is derived.
Abstract: Galerkin alternating-direction procedures are considered for the nonlinear parabolic equation \[ u_t - \sum_{i,j = 1}^N {\frac{\partial } {{\partial \xi _i }}} \left( {\tilde a_{ij} (\xi ,u)\frac{{\partial u}} {{\partial \xi_{j} }}} \right) + \sum_{i = 1}^N {\tilde b_i (\xi ,u)\frac{{\partial u}} {{\partial \xi _i }}} = f(\xi ,t,u), \] which is given on a nonrectangular, curved region. We use isoparametric elements and an approximation to the Jacobian of the isoparametric map which is based on patches of finite elements. With these procedures, a multidimensional problem can be solved as a series of one-dimensional problems. Optimal order $L^2$- and $H^1$-error estimates as well as optimal order work estimates are derived.
TL;DR: Exponentially fitted composite, multiderivative linear multistep methods of orders up to 5 are derived in this article, and a numerical investigation indicates that they may be A-stable for all choices of the fitting parameters.
Abstract: Exponentially fitted composite, multiderivative linear multistep methods of orders up to 5 are derived. A numerical investigation of these formulae indicates that they may be A-stable for all choices of the fitting parameters. An important aspect of the class of integration formulae considered is that each formula contains a “built-in” local error estimate—a facility lacking in most other exponentially fitted formulae. Some preliminary numerical results are given and these show that, for problems for which exponential fitting is appropriate, the new formulae provide a large saving in computational effort compared with conventional ones.
TL;DR: In this article, an iterative technique for finding the algebraically smallest (or largest) eigenvalue of the generalized eigen value problem, where A and M are real, symmetric, and M is positive definite, is discussed.
Abstract: In this paper we discuss an iterative technique for finding the algebraically smallest (or largest) eigenvalue of the generalized eigenvalue problem $A - \lambda M$, where A and M are real, symmetric, and M is positive definite. We assume that A and M are such that it is undesirable to factor the matrix $A - \sigma M$ for any value of $\sigma $. We prove that the algorithm is globally convergent, and that convergence is asymptotically quadratic. Finally, we discuss the modifications required in the algorithm to make it computationally feasible.
TL;DR: In this paper, it is shown how one can bound a posteriori the error made in the numerical solution of the TPBVP, including the truncation and the roundoff error.
Abstract: Consider a general two-point boundary value problem (TPBVP): \[ \begin{gathered} y'(t) = f(t,y), \hfill \\ B_1 y(a) + B_2 y(b) = w,\quad \hfill \\ \end{gathered} a \leqq t \leqq b, \] where $f:R^{n + 1} \to R^n ,f \in C^2 ,B_1 $ and $B_2 $ are $n \times n$ matrices and $w \in R^n $.It is shown how one can bound a posteriori the error made in the numerical solution of the TPBVP. The error bounds obtained are rigorous and include the truncation and the roundoff error. In addition, the computations establish the existence of solutions to the TPBVP.Numerical schemes are developed for the case where $f(t,y)$ is a polynomial in t and y. Examples are given of computational existence proofs for problems where analytical existence proofs are not known.
TL;DR: In this paper, a new derivation and motivation of Laguerre's method is given, which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized Hermite or hyperosculatory interpolation.
Abstract: Laguerre’s method is an efficient and reliable method for finding zeros of polynomials and certain other functions. A new derivation and motivation of Laguerre’s method is given, which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized Hermite or hyperosculatory interpolation. Members of this new class share with Laguerre’s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized Hermite interpolation. Methods of order 4 and 3.303 are investigated and numerical results indicate that for large $(100 \times 100)$ eigenvalue problems the method of order 3.303 is as efficient and reliable as Laguerre’s method.
TL;DR: In this article, the convergence and rate-of-convergence results for the weighted Galerkin method of numerical solution of Cauchy type singular integral equations with regular kernels are shown to be identical to the corresponding method based on the reduction of the Cauche type singular equation to an equivalent Fredholm integral equation of the second kind.
Abstract: The well-known weighted Galerkin method for the direct numerical solution of Cauchy type singular integral equations of the first and the second kind, based on the approximation of the unknown function by a finite series of appropriate orthogonal polynomials, is shown to be identical to the corresponding method based on the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind and the application to the latter of the weighted Galerkin method. This result permits the direct transfer of the whole set of convergence and rate-of-convergence results for the weighted Galerkin method of numerical solution of Fredholm integral equations with regular kernels to the case of Cauchy type singular integral equations. An application to the case where Chebyshev polynomials are used is also made, and the available convergence results for the direct weighted Galerkin method are rederived with essentially no analysis.