Orthogonal spline collocation methods for biharmonic problems

Orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square. Particular attention is given to the Dirichlet biharmonic problem which is solved using capacitance matrix techniques. Received November 11, 1996     

Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles

Summary. Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally stable and of optimal order accuracy in the and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms. Received November 7, 1994 / Revised version received April 29, 1996     

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems. Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001     

A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension

Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown for a nonlinear parabolic initial-boundary value problem. Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999     

Corrected collocation methods for periodic pseudodifferential equations

Summary. We propose an approximation method for periodic pseudodifferential equations, which yields higher convergence rates in Sobolev spaces with negative order than the collocation method. The main idea consists in correcting the usual collocation solution in a certain way by the solution of a small Galerkin system for the same equation. Both trigonometric and spline approximation methods are considered. In most of the cases our convergence result even improves that of the qualocation method. Received January 3, 1994 / Revised version received August 17, 1994     

Multilevel additive and multiplicative methods for orthogonal spline collocation problems

Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Received March 1, 1994 / Revised version received January 23, 1996     

Qualocation methods for elliptic boundary integral equations

Summary. The qualocation methods developed in this paper, with spline trial and test spaces, are suitable for classes of boundary integral equations with convolutional principal part, on smooth closed curves in the plane. Some of the methods are suitable for all strongly elliptic equations; that is, for equations in which the even symbol part of the operator dominates. Other methods are suitable when the odd part dominates. Received December 27, 1996 / Revised version received April 14, 1997     

Spectral methods for initial boundary value problems with nonsmooth data

Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution. Received August 1, 1995 / Revised version received August 19, 1997     

Spline qualocation methods for variable-coefficient elliptic equations on curves

Summary. Here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type. The analysis exploits localization in the form of frozen coefficients, pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators, a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders. Numerical examples show the predicted high orders of convergence. Received January 29, 1998 / Published online: June 29, 1999     

Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit

Summary. In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21,23]. We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved. Received July 10, 2001 / Revised version received October 12, 2001 / Published online January 30, 2002     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998     

On the asymptotic stability properties of Runge-Kutta methods for delay differential equations

Summary. In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered with respect to the following test equation: where and is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent of the specific value of ). In this work we direct attention at the dependence of stability regions on a fixed delay . In particular, natural Runge-Kutta methods for DDEs are extensively examined. Received April 15, 1996 / Revised version received August 8, 1996     

Spline collocation for convolutional parabolic boundary integral equations

Summary. We consider spline collocation methods for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover for example the case of the single layer heat operator equation when the spatial domain is a disc. Received December 15, 1997 / Revised version received November 16, 1998 / Published online September 24, 1999     

The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

On one approach to a posteriori error estimates for evolution problems solved by the method of lines

Summary. In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the -norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented. Received September 2, 1999 / Revised version received March 6, 2000 / Published online March 20, 2001     

Preconditioning cubic spline collocation discretizations of elliptic equations

Summary. This work considers the uniformly elliptic operator defined by in (the unit square) with boundary conditions: on and on and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix . We discuss the condition numbers and the distribution of -singular values of the preconditioned matrices where is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator given by in with boundary conditions: on on . The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by Gauss points or the space of continuous functions which are linear on the triangles of the triangulation of using the Gauss points. When we obtain results on the eigenvalues of . In the general case we obtain bounds and clustering results on the -singular values of . These results are related to the results of Manteuffel and Parter [MP], Parter and Wong [PW], and Wong [W] for finite element discretizations as well as the results of Parter and Rothman [PR] for discretizations based on Legendre Spectral Collocation. Received January 1, 1994 / Revised version received February 7, 1995     

Symmetric collocation methods for linear differential-algebraic boundary value problems

Summary. We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally by index reduction techniques) and apply Gau?-type (for the differential part) and Lobatto-type (for the algebraic part) collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions, we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence when using the combination of Gau? and Lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods. Received September 22, 2000 / Revised version received February 7, 2001 / Published online August 17, 2001     

Fast algorithms for high-order spline collocation systems

Summary. In this paper, we present a complete eigenvalue analysis for arbitrary order -spline collocation methods applied to the Poisson equation on a rectangular domain with Dirichlet boundary conditions. Based on this analysis, we develop some fast algorithms for solving a class of high-order spline collocation systems which arise from discretizing the Poisson equation. Received April 8, 1997 / Revised version received August 29, 1997     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Finite difference preconditioning cubic spline collocation method of elliptic equations

Summary. We discuss a finite difference preconditioner for the interpolatory cubic spline collocation method for a uniformly elliptic operator defined by in (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix where is the matrix of the collocation discretization operator corresponding to , and is the matrix of the finite difference operator corresponding to the uniformly elliptic operator given by in with homogeneous Dirichlet boundary conditions. Finally we mention a bound of -singular values of for a general elliptic operator in . Received December 11, 1995 / Revised version received June 20, 1996