Showing papers in "Mathematics of Computation in 1991"

Spectral Methods in Fluid Dynamics.

Abstract: 1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat Equation.- 1.2.3. A Legendre Tau Method for the Poisson Equation.- 1.2.4. Basic Aspects of Galerkin, Tau and Collocation Methods.- 1.3. The Equations of Fluid Dynamics.- 1.3.1. Compressible Navier-Stokes.- 1.3.2. Compressible Euler.- 1.3.3. Compressible Potential.- 1.3.4. Incompressible Flow.- 1.3.5. Boundary Layer.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.1.1. The Continuous Fourier Expansion.- 2.1.2. The Discrete Fourier Expansion.- 2.1.3. Differentiation.- 2.1.4. The Gibbs Phenomenon.- 2.2. Orthogonal Polynomials in ( - 1, 1).- 2.2.1. Sturm-Liouville Problems.- 2.2.2. Orthogonal Systems of Polynomials.- 2.2.3. Gauss-Type Quadratures and Discrete Polynomial Transforms.- 2.3. Legendre Polynomials.- 2.3.1. Basic Formulas.- 2.3.2. Differentiation.- 2.4. Chebyshev Polynomials.- 2.4.1. Basic Formulas.- 2.4.2. Differentiation.- 2.5. Generalizations.- 2.5.1. Jacobi Polynomials.- 2.5.2. Mapping.- 2.5.3. Semi-Infinite Intervals.- 2.5.4. Infinite Intervals.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.1.1. Fourier Galerkin.- 3.1.2. Fourier Collocation.- 3.1.3. Chebyshev Tau.- 3.1.4. Chebyshev Collocation.- 3.2. Convolution Sums.- 3.2.1. Pseudospectral Transform Methods.- 3 2 2 Aliasing Removal by Padding or Truncation.- 3.2.3. Aliasing Removal by Phase Shifts.- 3.2.4. Convolution Sums in Chebyshev Methods.- 3.2.5. Relation Between Collocation and Pseudospectral Methods.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.4.1. Polar Coordinates.- 3.4.2. Spherical Polar Coordinates.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.2.1. The First-Derivative Operator.- 4.2.2. The Second-Derivative Operator.- 4.3. Some Standard Schemes.- 4.3.1. Multistep Schemes.- 4.3.2. Runge-Kutta Methods.- 4.4. Special Purpose Schemes.- 4.4.1. High Resolution Temporal Schemes.- 4.4.2. Special Integration Techniques.- 4.4.3. Lerat Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.1.1. Fourier Approximations.- 5.1.2. Chebyshev Tau Approximations.- 5.1.3. Schur-Decomposition and Matrix-Diagonalization.- 5.2. Fundamentals of Iterative Methods.- 5.2.1. Richardson Iteration.- 5.2.2. Preconditioning.- 5.2.3. Non-Periodic Problems.- 5.2.4. Finite-Element Preconditioning.- 5.3. Conventional Iterative Methods.- 5.3.1. Descent Methods for Symmetric, Positive-Definite Systems.- 5.3.2. Descent Methods for Non-Symmetric Problems.- 5.3.3. Chebyshev Acceleration.- 5.4. Multidimensional Preconditioning.- 5.4.1. Finite-Difference Solvers.- 5.4.2. Modified Finite-Difference Preconditioners.- 5.5. Spectral Multigrid Methods.- 5.5.1. Model Problem Discussion.- 5.5.2. Two-Dimensional Problems.- 5.5.3. Interpolation Operators.- 5.5.4. Coarse-Grid Operators.- 5.5.5. Relaxation Schemes.- 5.6. A Semi-Implicit Method for the Navier-Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier-Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.2.1. A Spectral Galerkin Solution Technique.- 7.2.2. Treatment of the Nonlinear Terms.- 7.2.3. Refinements.- 7.2.4. Pseudospectral and Collocation Methods.- 7.3. Inhomogeneous Flows.- 7.3.1. Coupled Methods.- 7.3.2. Splitting Methods.- 7.3.3. Galerkin Methods.- 7.3.4. Other Confined Flows.- 7.3.5. Unbounded Flows.- 7.3.6. Aliasing in Transition Calculations.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.4.1. Choice of Mesh.- 7.4.2. Coupled Methods.- 7.4.3. Splitting Methods.- 7.4.4. Other Methods.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.5.1. Potential Flow.- 8.5.2. Ringleb Flow.- 8.5.3. Astrophysical Nozzle.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. Fourier Approximation.- 9.1.1. Inverse Inequalities for Trigonometric Polynomials.- 9.1.2. Estimates for the Truncation and Best Approximation Errors.- 9.1.3. Estimates for the Interpolation Error.- 9.2. Sturm-Liouville Expansions.- 9.2.1. Regular Sturm-Liouville Problems.- 9.2.2. Singular Sturm-Liouville Problems.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.4.1. Inverse Inequalities for Algebraic Polynomials.- 9.4.2. Estimates for the Truncation and Best Approximation Errors.- 9.4.3. Estimates for the Interpolation Error.- 9.5. Chebyshev Approximations.- 9.5.1. Inverse Inequalities for Polynomials.- 9.5.2. Estimates for the Truncation and Best Approximation Errors.- 9.5.3. Estimates for the Interpolation Error.- 9.5.4. Proofs of Some Approximation Results.- 9.6. Other Polynomial Approximations.- 9.6.1. Jacobi Polynomials.- 9.6.2. Laguerre and Hermite Polynomials.- 9.7. Approximation Results in Several Dimensions.- 9.7.1. Fourier Approximations.- 9.7.2. Legendre Approximations.- 9.7.3. Chebyshev Approximations.- 9.7.4. Blended Fourier and Chebyshev Approximations.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.1.1. A Fourier Galerkin Method for the Wave Equation.- 10.1.2. A Chebyshev Collocation Method for the Heat Equation.- 10.1.3. A Legendre Tau Method for the Poisson Equation.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.4.1. Galerkin Methods.- 10.4.2. Tau Methods.- 10.4.3. Collocation Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.5.1. Conditions for Stability and Convergence: The Parabolic Case.- 10.5.2. Conditions for Stability and Convergence: The Hyperbolic Case.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.1.1. Legendre Methods.- 11.1.2. Chebyshev Methods.- 11.1.3. Other Boundary Value Problems.- 11.2. Advection-Diffusion Equation.- 11.2.1. Linear Advection-Diffusion Equation.- 11.2.2. Steady Burgers Equation.- 11.3. Navier-Stokes Equations.- 11.3.1. Compatibility Conditions Between Velocity and Pressure.- 11.3.2. Direct Discretization of the Continuity Equation: The \"inf-sup\" Condition.- 11.3.3. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser-Schumann Method.- 11.3.4. Navier-Stokes Equations in Streamfunction Formulation.- 11.4. The Eigenvalues of Some Spectral Operators.- 11.4.1. The Discrete Eigenvalues for Lu = ? uxx.- 11.4.2. The Discrete Eigenvalues for Lu = ? vuxx + bux.- 11.4.3. The Discrete Eigenvalues for Lu = ux.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.1.1. Periodic Boundary Conditions.- 12.1.2. Non-Periodic Boundary Conditions.- 12.1.3. Hyperbolic Systems.- 12.1.4. Spectral Accuracy for Non-Smooth Solutions.- 12.2. Heat Equation.- 12.2.1. Semi-Discrete Approximation.- 12.2.2. Fully Discrete Approximation.- 12.3. Advection-Diffusion Equation.- 12.3.1. Semi-Discrete Approximation.- 12.3.2. Fully Discrete Approximation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.2.1. Notation.- 13.2.2. Discretization.- 13.2.3. Solution Techniques.- 13.2.4. Examples.- 13.3. Variational Methods.- 13.3.1. Formulation.- 13.3.2. The Spectral-Element Method.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.5.1. Patching Methods.- 13.5.2. Equivalence Between Patching and Variational Methods.- 13.6. Some Stability and Convergence Results.- 13.6.1. Patching Methods.- 13.6.2. Variational Methods.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi-Gauss-Lobatto Roots.- References.

Nonlinear optimization: complexity issues

Abstract: Optimization and convexity complexity theory convex quadratic programming non-convex quadratic programming local optimization complexity in the black-box model.

On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics

Abstract: The authors consider the equations of stationary, incompressible magneto-hydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. They discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

Some Estimates for a Weighted L 2 Projection

Abstract: This paper is devoted to the error estimates for some weighted L2 projections. Nearly optimal estimates are obtained. These estimates can be applied to the analysis of the usual multigrid method, multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities.

Convergence estimates for multigrid algorithms without regularity assumptions

Abstract: A new technique for proving rate of convergence estimates of multi- grid algorithms for symmetric positive definite problems will be given in this paper. The standard multigrid theory requires a "regularity and approxima- tion" assumption. In contrast, the new theory requires only an easily verified approximation assumption. This leads to convergence results for multigrid re- finement applications, problems with irregular coefficients, and problems whose coefficients have large jumps. In addition, the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications.

Convergence estimates for product iterative methods with applications to domain decomposition

Abstract: In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space % which are defined in terms of products of operators defined with respect to a number of subspaces. The simplest algorithm of this sort has an error-reducing operator which is the product of orthogonal projections onto the complement of the subspaces. New normreduction estimates for these iterative techniques will be presented in an abstract setting. Applications are given for overlapping Schwarz algorithms with many subregions for finite element approximation of second-order elliptic problems.

Polytope volume computation

Abstract: A combinatorial form of Gram's relation for convex polytopes can be adapted for use in computing polytope volume. We present an algorithm for volume computation based on this observation. This algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities, P = {x E R': Ax < b} . As an illustration we compute a formula for the volume of a projective image of the n-cube. From this formula we deduce that, when A and b have rational entries (so that the volume of P is also a rational number), the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of A and b . This settles a question posed by Dyer and Frieze.

The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms

Abstract: We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the \"V cycle, W cycle and a \"V cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way. We apply this theory to various numerical approximations of second-order elliptic boundary value problems. In our first example, we consider certain finite difference multigrid algorithms. In the second example, we consider a finite element multigrid algorithm with nested spaces, which however uses a prolongation operator that does not coincide with the natural subspace imbedding. The third example gives a multigrid algorithm derived from a loosely coupled sequence of approximation grids. Such a loosely coupled grid structure results from the most natural standard finite element application on a domain with curved boundary. The fourth example develops and analyzes a multigrid algorithm for a mixed finite element method using the so-called Raviart-Thomas elements.

Spline Models for Observational Data.

Abstract: Foreword 1. Background 2. More splines 3. Equivalence and perpendicularity, or, what's so special about splines? 4. Estimating the smoothing parameter 5. 'Confidence intervals' 6. Partial spline models 7. Finite dimensional approximating subspaces 8. Fredholm integral equations of the first kind 9. Further nonlinear generalizations 10. Additive and interaction splines 11. Numerical methods 12. Special topics Bibliography Author index.

Nonconforming finite element methods for the equations of linear elasticity

Abstract: In the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions, the main difficulty in the analysis is to prove that an appropriate discrete version of Korn's second inequality is valid. Such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears. Optimal-order error estimates, uniform for Poisson ratio v E [0, 1/2), are then derived for the corresponding P2 and P3 methods. This contrasts with the use of C finite elements, where there is a deterioration in the convergence rate as v -1/2 for piecewise polynomials of degree < 3. Modifications of the continuous methods and the nonconforming linear method which also give uniform optimal-order error estimates are discussed.

On L[1]-approximation

Abstract: Preface 1. Preliminaries 2. Approximation from finite-dimensional subspaces of L1 3. Approximation from finite-dimensional subspaces in C1 (K, ) 4. Unicity subspaces and property A 5. One-sided L1-approximation 6. Discrete lm1 - approximation 7. Algorithms Appendices References Author index Subject index.

Approximation of some diffusion evolution equations in unbounded domains by Hermite functions

Abstract: Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.

NURBS for Curve and Surface Design

Abstract: From the Publisher: NURBS (nonuniform rational B-splines) promises to be the future geometry standard for free-form curves and surfaces. This volume contains recent results and new NURBS techniques and developments--the most important being curve/surface from the CAD/CAM industry. The book has been carefully refereed. Researchers, system programmers, and graduate students in CAGD, CAD/CAM and computer graphics will find this book uniquely suited to their field of work.

Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls

Abstract: We examine certain analytic and numerical aspects of optimal control problems for the stationary Navier-Stokes equations. The controls considered may be of either the distributed or Neumann type; the functionals minimized are either the viscous dissipation or the L4_distance of candidate flows to some desired flow. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider the approximation, by finite element methods, of solutions of the optimality system and derive optimal error estimates.

Finding isomorphisms between finite fields

Abstract: We show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time. Every finite field has cardinality p" for some prime number p and some positive integer « . Conversely, if p is a prime number and « a positive integer, then there exists a field of cardinality p", and any two fields of cardinality p" are isomorphic. These results are due to E. H. Moore (1893) (10). In the present paper we are interested in an algorithmic version of his theorem, in particular of the uniqueness part. We say that a finite field is explicitly given if, for some basis of the field over its prime field, we know the product of any two basis elements, expressed in the same basis. Let, more precisely, p be a prime number and « a positive integer. Then by explicit data for a finite field of cardinality p" we mean a system of « elements (ajjk)" k=x of the prime field Fp = Z/pZ suchthat F^ becomes a field with the ordinary addition and multiplication by elements of Fp , and the multiplication determined by n

Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis

Abstract: A finite difference technique on rectangular cell-centered grids with local refinement is proposed in order to derive discretizations of second-order elliptic equations of divergence type approximating the so-called balance equa1 1/2 tion. Error estimates in a discrete H -norm are derived of order h ' for a simple symmetric scheme, and of order h ' for both a nonsymmetric and a more accurate symmetric one, provided that the solution belongs to H +a for a > \\ and a > \\ , respectively.

Discrete least squares approximation by trigonometric polynomials

Abstract: We present an efficient and reliable algorithm for discrete least squares approximation of a real-valued function given at arbitrary distinct nodes in [0, 2tt) by trigonometric polynomials. The algorithm is based on a scheme for the solution of an inverse eigenproblem for unitary Hessenberg matrices, and requires only O(mn) arithmetic operations as compared with 0(mn ) operations needed for algorithms that ignore the structure of the problem. Moreover, the proposed algorithm produces consistently accurate results that are often better than those obtained by general QR decomposition methods for the least squares problem. Our algorithm can also be used for discrete least squares approximation on the unit circle by algebraic polynomials.

Real-Time Integration Methods for Mechanical System Simulation

Abstract: This study focuses on the mechanical system dynamics applications of differential-algebraic equations (DAE). Selected papers develop the foundations of DAE as they appear in mechanical system dynamics applications, and explore the foundations of contemporary methods for the numerical solution of DAE in mechanical system dynamics. Many theoretical methods and methods of generalized co-ordinate partitioning for the numerical integration of DAE are examined. The text has been designed to encompass a broad range of applications in engineering and applied physics.

Vortex methods and vortex motion

Abstract: Vortex phenomena in fluid flows and the experimental, theoretical, and numerical methods used to characterize them are discussed in reviews by leading experts. Chapters are devoted to an overview of vortex methods, the convergence of vortex methods, graphic displays from numerical simulations, physical vortex visualizations, the four principles of vortex motion, the visualization and computation of hovering-mode vortex dynamics, turbulence and vortices in superfluid He, and statistical-mechanics approaches to vortices and turbulence. Extensive photographs and sample computer graphics are provided.

A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions

Abstract: We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation parameter, of order h1/2 in a global energy norm which is stronger than the L2 norm. This order is optimal in this norm for our choice of trial functions.

A classification of the cosets of the Reed-Muller code R(1, 6)

Abstract: The weight distribution of a coset of a Reed-Muller code M( 1, m) is invariant under a large transformation group consisting of all affine rearrangements of a vector space with dimension m . We discuss a general algorithm that produces an ordered list of orbit representatives for this group action. As a byproduct the procedure finds the order of the symmetry group of a coset. With m = 6 we can implement the algorithm on a computer and find that there are 150357 equivalence classes. These classes produce 2082 distinct weight distributions. Their symmetry groups have 122 different orders.

An automatic quadrature for Cauchy principal value integrals

Abstract: An automatic quadrature is presented for computing Cauchy principal value integrals Q(f; c) = Faf(t)/(t c) dt, a < c < b, for smooth functions f(t) . After subtracting out the singularity, we approximate the function f(t) by a sum of Chebyshev polynomials whose coefficients are computed using the FTT. The evaluations of Q(f; c) for a set of values of c in (a, b) are efficiently accomplished with the same number of function evaluations. Numerical examples are also given.

Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem

Abstract: Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational prob- lems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gra- dient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.

Solving solvable quintics

Abstract: 5 3 2 Abstract. Let f{x) = x +px +qx +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if f(x) is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group F20 of order 20 in the symmetric group S5). When f(x) is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order.

The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs

Abstract: Consider solving the Dirichlet problem Au(P) = o, Pem2\s, u(P) = h(P), Pes, sup \u(P)\ < CO, Pes2 with S a smooth open curve in the plane. We use single-layer potentials to construct a solution u(P). This leads to the solution of equations of the form j g(Q)\og\P-Q\dS(Q) = h(P), PeS. This equation is reformulated using a special change of variable, leading to a new first-kind equation with a smooth solution function. This new equation is split into a principal part, which is explicitly invertible, and a compact perturbation. Then a discrete Galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. A complete convergence analysis is given; numerical examples conclude the paper. with S a smooth open contour in the plane. This equation arises in a variety of contexts, one being the study of elasticity crack problems in the plane. We propose a new numerical method for solving (1.1), and then use it to solve some potential theory problems in the plane. The method takes account of the expected singularities in g at the ends of the contour in an entirely natural way, and the method is shown to converge rapidly when the curve 5" and the data are sufficiently smooth. We limit the functions g and h to be real, although the following development extends easily to the complex case. Let S have a parametrization

An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates

Abstract: A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. A typical triangula- tion is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the dis- crete interface so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors be- tween consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in LP -based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods — 3/2 —2 on quasi-uniform meshes, namely 0(r ) as opposed to 0(x ), to achieve the same global asymptotic accuracy; here r > 0 is the (uniform) time step. 112 A rate of convergence of essentially 0(x ' ) is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0(x~ ' ), which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme's efficiency in approximating both solutions and interfaces.

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems.

Abstract: The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Chebyshev-Vandermonde systems

Abstract: A Chebyshev-Vandermonde matrix is obtained by replacing the monomial entries of a Vandermonde matrix by Chebyshev polynomials /> for an ellipse. The ellipse is also allowed to be a disk or an interval. We present a progressive scheme for allocating distinct nodes zk on the boundary of the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. Fast progressive algorithms for the solution of the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. We show that the node allocation is such that the solution computed by the progressive algorithms is fairly insensitive to perturbations in the right-hand side vector. Computed examples illustrate the numerical behavior of the schemes. Our analysis can also be used to bound the condition number of the polynomial interpolation operator defined by Newton's interpolation formula. This extends earlier results of Fischer and the first author.