Showing papers in "SIAM Journal on Numerical Analysis in 1989"
TL;DR: In this paper, the convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed, and it is shown that the convergence may (and in fact in some cases must) fail, with or without postprocessing of the numerical solution.
Abstract: The convergence of the Fourier method for scalar nonlinear conservation laws which exhibit spontaneous shock discontinuities is discussed. Numerical tests indicate that the convergence may (and in fact in some cases must) fail, with or without post-processing of the numerical solution. Instead, a new kind of spectrally accurate vanishing viscosity is introduced to augment the Fourier approximation of such nonlinear conservation laws. Using compensated compactness arguments, it is shown that this spectral viscosity prevents oscillations, and convergence to the unique entropy solution follows.
432 citations
TL;DR: In this article, a non-linear Galerkin method is proposed to integrate evolution differential equations on a nonlinear manifold, which is well adapted to the long-term integration of such equations.
Abstract: This article presents a new method of integrating evolution differential equations—the non-linear Galerkin method—that is well adapted to the long-term integration of such equations.While the usual Galerkin method can be interpreted as a projection of the considered equation on a linear space, the methods considered here are related to the projection of the equation on a nonlinear manifold. From the practical point of view some terms have been identified as small, and sometimes.(but not always) disregarded.
366 citations
TL;DR: The BFGS method for unconstrained optimization, using a variety of line searches, including backtracking, is shown to be globally and superlinearly convergent on uniformly convex problems.
Abstract: The BFGS update formula is shown to have an important property that is inde- pendent of the algorithmic context of the update, and that is relevant to both constrained and unconstrained optimization. The BFGS method for unconstrained optimization, using a variety of line searches, including backtracking, is shown to be globally and superlinearly convergent on uniformly convex problems. The analysis is particularly simple due to the use of some new tools introduced in this paper.
339 citations
TL;DR: In this article, a simple finite element method for the Reissner-Mindlin plate model in the primitive variables is presented and analyzed, which uses nonconforming linear finite elements for the transverse displacement and conforming linear infinite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging.
Abstract: A simple finite element method for the Reissner–Mindlin plate model in the prim-itive variables is presented and analyzed The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging It is proved that the method converges with optimal order uniformly with respect to thickness
275 citations
TL;DR: A number of well-known optimal interpolation results are generalized in the case where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains.
Abstract: This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.
235 citations
TL;DR: A class of methods is introduced that generalizes the so-called exponential fitting method to two-dimensional problems and, more specifically, to equations of the type ${\operatorname{div} (\underset0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{
abla } \psi) through the change of variables.
Abstract: A class of methods is introduced that generalizes the so-called exponential fitting method to two-dimensional problems and, more specifically, to equations of the type ${\operatorname{div}} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{
abla } u + u\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{
abla } \psi ) = f$, with $\psi $ given and $|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{
abla } \psi |$ big in a part of the domain. The basic idea is the following: (i) write the equation as ${\operatorname{div}} (e^{ - \psi } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{
abla } \rho ) = f$ through the change of variables $u = \rho e^{ - \psi } $; (ii) discretize the symmetric equation by means of a method that uses some kind of harmonic average for the coefficient $e^{ - \psi } $ (mixed, hybrid, etc.); (iii) write the discrete scheme in terms of the nodal values of $u = e^{ - \psi } \rho $. This produces an M-matrix.
164 citations
TL;DR: In this paper, it was shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation.
Abstract: It is shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation. The main conditions for this result are that the Cauchy problem for the Boltzmann equation has a sufficiently smooth solution and that the discretization parameters (cell size, timestep and test particle number) satisfy suitable constraints.
156 citations
TL;DR: In this article, a pseudospectral explicit scheme for solving linear, periodic, parabolic problems is described, which has infinite accuracy both in time and in space and is achieved while the time resolution parameter $M(M = O({1 / {\Delta t}})$ for time marching algorithm) and the space resolution parameter N(N = O(1/ {\Delta x))}}$ must satisfy the common stability condition, which must be satisfied in any explicit finite order time algorithm.
Abstract: A pseudospectral explicit scheme for solving linear, periodic, parabolic problems is described. It has infinite accuracy both in time and in space. The high accuracy is achieved while the time resolution parameter $M(M = O({1 / {\Delta t}})$ for time marching algorithm) and the space resolution parameter $N(N = O({1 / {\Delta x))}}$ must satisfy $M = O(N^{1 + \varepsilon } )\varepsilon > 0$, compared to the common stability condition $M = O(N^2 )$, which must be satisfied in any explicit finite-order time algorithm.
135 citations
TL;DR: In this paper, a continuous in-time finite-element Galerkin approximation is considered and the nonconforming Morley element is used to derive optimal order error bounds in $L^2 $.
Abstract: The Cahn–Hilliard equation is a nonlinear evolutionary equation that is fourth order in space. In this paper a continuous in-time finite-element Galerkin approximation is considered. We use the nonconforming Morley element and derive optimal order error bounds in $L^2 $.
131 citations
TL;DR: In this paper, improved error estimates for a finite-element modified method of characteristics for a coupled system of partial differential equations modeling flow in porous media were derived for a porous media.
Abstract: Improved error estimates are derived for a finite-element modified method of characteristics for a coupled system of partial differential equations modeling flow in porous media.These results impro...
129 citations
TL;DR: The cheater's homotopy as mentioned in this paper is a continuation method that follows paths to all solutions in a polynomial system with varying coefficients, and it can be found with an amount of computational work roughly proportional to the actual number of solutions.
Abstract: A procedure is introduced for solving systems of polynomial equations that need to be solved repetitively with varying coefficients. The procedure is based on the cheater’s homotopy, a continuation method that follows paths to all solutions. All solutions are found with an amount of computational work roughly proportional to the actual number of solutions. Previous general methods normally require an amount of computation roughly proportional to the total degree.
TL;DR: This paper rewrites some of the conditions on the Hermite derivatives that are sufficient for a piecewise bicubic function to be monotonic and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant.
Abstract: This paper describes an algorithm for monotone interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. In [SIAM J. Numer. Anal. 22 (1985), pp. 386–400] the authors developed conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. The present paper rewrites some of these conditions and presents a much simpler five-step algorithm for satisfying them that produces a visually pleasing monotone interpolant. The result of the algorithm does not depend on the order of the independent variables nor on whether the inequalities are swept left-to-right or right-to-left.
TL;DR: In this paper, implicit Runge-Kutta-Nystrom (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution.
Abstract: Implicit Runge–Kutta–Nystrom (RKN) methods are constructed for the integration of second-order differential equations possessing an oscillatory solution. Based on a linear homogeneous test model we analyse the phase errors (or dispersion) introduced by these methods and derive so-called dispersion relations. Diagonally implicit RKN methods of relatively low algebraic order are constructed, which have a high order of dispersion (up to 10). Application of these methods to a number of test examples (linear as well as nonlinear) yields a greatly reduced phase error when compared with “conventional” DIRKN methods.
TL;DR: In this paper, the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers' equation is analyzed, and it is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth.
Abstract: The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.
TL;DR: It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used.
Abstract: It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used. Bounds for the Euclidean condition number as a function of the number of degrees f freedom are derived in $n \geq 2$ dimensions. When $n \geq 3$ the bound is the same as for the regular mesh case, but when $n = 2$ a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes. Applications of the results to the conjugate-gradient iterative method for solving such linear systems are given.
TL;DR: In this paper, a general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix' blocks, and optimal order $L 2 $-error estimates are derived.
Abstract: A general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix' blocks. The Warren and Root model is shown to be a crude approximation to this model. The general model consists of a parabolic equation coupled to a series of parabolic equations. It is shown that the coupling term can be viewed as a positive-semidefinite perturbation of the time derivative, and hence it is verified that the model is well posed. A finite element method is presented to approximate the solution, and optimal order $L^2 $-error estimates are derived.
TL;DR: In this paper, error estimates for finite-element approximations of the solutions to semilinear parabolic problems are proved under the hypothesis that the exact solution is asymptotically stable as $t \to \infty $.
Abstract: Error estimates for finite-element approximations of the solutions to semilinear parabolic problems are proved. Under the hypothesis that the exact solution is asymptotically stable as $t \to \infty $, error estimates of optimal order that hold uniformly on the unbounded time interval $0 \leqq t < \infty $ are obtained. Both semidiscrete and completely discrete approximations are considered.
TL;DR: In this paper, the weighted Sobolev spaces on a square, whose weight is a given power of the product of the distances to the edges, are introduced and trace theorems related to these spaces are proved, then a regularity result for the Dirichlet problem for the Laplace operator is proved.
Abstract: The weighted Sobolev spaces on a square, whose weight is a given power of the product of the distances to the edges, are introduced. Trace theorems related to these spaces are proved, then a regularity result for the Dirichlet problem for the Laplace operator is proved. Next several projection operators with polynomial values are considered for which approximation results in weighted norms are stated. Finally, a collocation spectral method for the Dirichlet problem for the Laplace operator with inhomogeneous boundary conditions is analyzed. This work includes new results on Chebyshev approximation.
TL;DR: An optimal-order multigrid method for solving the biharmonic equation using Morley nonconforming finite elements is developed.
Abstract: An optimal-order multigrid method for solving the biharmonic equation using Morley nonconforming finite elements is developed.
TL;DR: The convergence behaviour of a class of iterative methods for solving the constrained minimization problem is analysed and it is found that they are sufficiently general to ensure global convergence of the iterates to the solution of the problem at an asymptotic (two-step Q-) superlinear rate.
Abstract: The convergence behaviour of a class of iterative methods for solving the constrained minimization problem is analysed. The methods are based on the sequential minimization of a simple differentiable penalty function. They are sufficiently general to ensure global convergence of the iterates to the solution of the problem at an asymptotic (two-step Q-) superlinear rate.
TL;DR: Estimates for the errors incurred for a “semi-discrete” approximation to the underlying Vlasov–Poisson system are given by first superimposing a rectangular grid or mesh on all of phase space and then replacing the initial continuous distribution of charges or masses by discrete charges or mass located at the centroid of each grid cell.
Abstract: For Vlasov–Poisson systems, particle methods are numerical techniques that simulate the behavior of a plasma by a large set of charged superparticles, which obey the classical laws of electrostatics The trajectories of these charged particles are then followed Estimates for the errors incurred for a “semi-discrete” approximation to the underlying Vlasov–Poisson system are given by first superimposing a rectangular grid or mesh on all of phase space and then replacing the initial continuous distribution of charges or masses by discrete charges or masses located at the centroid of each grid cell Our analysis, on one hand, generalizes that of G H Cottet and P A Raviart (SIAM J Numer Anal, 21 (1984), pp 52–76) to higher-dimensional Vlasov–Poisson systems, and, on the other, those fundamental results of Ole Hald (SIAM J Numer Anal, 16 (1979), pp 726–755) and of J T Beale and A Majda (Math Comp, 39 (1982), pp 1–52) on vortex methods for two- and three-dimensional Euler equations, to particl
TL;DR: In this paper, the authors studied the order, stability, and convergence properties of implicit Runge-Kutta (IRK) methods applied to differential/algebraic systems with index greater than one.
Abstract: This paper studies the order, stability, and convergence properties of implicit Runge–Kutta (IRK) methods applied to differential/algebraic systems with index greater than one. These methods do not in general attain the same order of accuracy for higher index differential/algebraic systems as they do for index 1 systems or for purely differential systems. Necessary and sufficient conditions on the method coefficients are derived to ensure that the local and global errors of the method attain a given order of accuracy for higher index linear constant coefficient systems. IRK methods applied to nonlinear semi-explicit index 2 systems are studied, and a sufficient set of conditions is derived which ensures that a method is accurate to a given order for these systems. Finally, some numerical experiments are presented that illustrate the theoretical results and demonstrate the effects of roundoff' errors on the solution.
TL;DR: In this article, the authors investigated the condition number appropriate to the componentwise backward error analysis of triangular systems and found that the conditioning of a triangular system depends on the right-hand side as well as the coefficient matrix.
Abstract: Triangular systems play a fundamental role in matrix computations. It has been prominently stated in the literature, but is perhaps not widely appreciated, that solutions to triangular systems are usually computed to high accuracy—higher than the traditional condition numbers for linear systems suggest. This phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems. Results of Wilkinson are unified and extended. Among the conclusions are that the conditioning of a triangular system depends on the right-hand side as well as the coefficient matrix; that use of pivoting in LU, QR, and Cholesky factorisations can greatly improve the conditioning of a resulting triangular system; and that a triangular matrix may be much more or less ill-conditioned than its transpose.
TL;DR: Here, further theoretical justification, including a global convergence theorem, is provided and modifications are suggested that allow the efficient implementation of the merit function while maintaining the important convergence properties.
Abstract: n a previous work [P. Boggs and J. Tolle, SIAM J. Numer. Anal., 21 (1984), pp. 1146–1161], the authors introduced a merit function for use with the sequential quadratic programming (SQP) algorithm for solving nonlinear programming problems. Here, further theoretical justification, including a global convergence theorem, is provided. In addition, modifications are suggested that allow the efficient implementation of the merit function while maintaining the important convergence properties. Numerical results are presented demonstrating the effectiveness of the procedure.
TL;DR: In this paper, the convergence of the point vortex method and the vortex blob method is demonstrated for vortex sheets with both spatial and temporal discretization and with simulated round-of-l error.
Abstract: Computation of the evolution of vortex sheets is delicate because of Kelvin–Helmholtz instability and singularity formation (infinite curvature). Convergence of the point vortex method and the vortex blob method is demonstrated for vortex sheets with both spatial and temporal discretization and with simulated roundofl error. The initial data is assumed to be a small analytic perturbation of a flat, uniform sheet. The proof works for a short time interval, certainly less than the first time of singularity formation. The analysis is performed in an analytic function space using the abstract Cauchy–Kowalewski Theorem. A numerical-analytic interpretation of analyticity is given.
TL;DR: Both the standard semidiscrete in space Galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations, and the Crank–Nicolson discretizations in time are considered as examples of completely discrete schemes.
Abstract: Some recent results concerning maximum-norm superconvergence of the gradient in piecewise linear finite-element approximations of an elliptic problem are carried over to a parabolic problem. Both the standard semidiscrete in space Galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations, and the Crank–Nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes.
TL;DR: In this paper, a mathematical method for studying discontinuous solutions for systems in non-conservative form is introduced, and the evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics.
Abstract: A (rigorous) mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form. The evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics.
TL;DR: In this article, the authors analyzed the Rayleigh Quotient Iteration (RQI) algorithm for symmetric matrices and characterized the sets of points for which the algorithm will converge to an eigenvector.
Abstract: This paper analyzes the Rayleigh Quotient Iteration algorithm for symmetric matrices. Dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector. It is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues.
TL;DR: In this article, the authors considered four error control strategies that can be used in methods based on Runge-Kutta formula pairs with interpolants and showed that a strategy based on direct defect control can provide significant advantages over existing strategies with only a modest increase in cost.
Abstract: There has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete Runge–Kutta methods. With the availability of these schemes it can now be asked that a numerical method provide a continuous approximation to the solution. Thispaper, rather than view such a continuous method as an interpolant superimposed on a standard discrete method, considers how the interpolant and its associated defect can be effectively used in the underlying error and stepsize control mechanism.In particular four error control strategies are considered that can be used in methods based on Runge–Kutta formula pairs with interpolants. An asymptotic and a nonasymptotic analysis of each strategy are presented. It is shown that a strategy based on direct defect control can provide significant advantages over existing strategies with only a modest increase in cost.
TL;DR: In this article, a point relaxation method for the liquid crystal problem is proposed and analyzed, and it is proved that the energy of successive iterates is nonincreasing for the point relaxation algorithm with relaxation parameter $0 < \omega < 2$ for a class of material constants.
Abstract: A point relaxation method for the liquid crystal problem is proposed and analyzed. The liquid crystal problem is a minimization problem with a nonconvex local constraint. It is proved that the energy of successive iterates is nonincreasing for the point relaxation method with relaxation parameter $\omega $ satisfying $0 < \omega < 2$ for a class of material constants. It is also shown that the difference between successive iterates converges to zero, and that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point and which satisfy the constraint. Numerical results are given which demonstrate the improved rate of convergence for overrelaxation.