Romberg quadrature using the Bulirsch sequence

Summary. In this note, we prove a conjecture of Bulirsch concerning the definiteness of the Romberg quadrature rules using the Bulirsch sequence. We compare these rules with the classical Romberg scheme and the Gaussian rules. Received May 16, 2000 / Published online May 30, 2001     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

Summary. We show that, if (), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type , , fulfills uniformly for all , and hence it is of optimal order of magnitude in the classes (). Here, is a weight function with the property . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restriction can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems. Received November 21, 1994     

Error bounds for minimal energy bivariate polynomial splines

Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002 RID="*" ID="*" Supported by the National Science Foundation under grant DMS-9870187 RID="**" ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160     

Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow

Summary. We consider the finite element approximation of a non-Newtonian flow, where the viscosity obeys a general law including the Carreau or power law. For sufficiently regular solutions we prove energy type error bounds for the velocity and pressure. These bounds improve on existing results in the literature. A key step in the analysis is to prove abstract error bounds initially in a quasi-norm, which naturally arises in degenerate problems of this type. Received May 25, 1993 / Revised version received January 11, 1994     

Construction of asymptotically best quadrature formulas

Summary. We show that, for integrals with arbitrary integrable weight functions, asymptotically best quadrature formulas with equidistant nodes can be obtained by applying a certain scheme of piecewise polynomial interpolation to the function to be integrated, and then integrating this interpolant. Received August 7, 1991     

Error indicators for the Navier-Stokes equations in stream function and vorticity formulation

This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second step, these results are extended to the Navier-Stokes equations. Received March 25, 1996 / Revised version received April 7, 1997     

Optimal $\mathcal{L}^2$ error bounds for MITC3 type element

Summary. In this paper, we derive the optimal error bounds for the stabilized MITC3 element [3], the MIN3 type element [7] and the T3BL element [8]. In this way we have solved the problem proposed recently in [5] in a positive manner. Moreover, we estimate the difference between stabilized MITC3 and MIN3 and show it is of order uniform in the plate thickness. Received May 31, 2000 / Revised version received April 2, 2001 / Published online September 19, 2001     

Gaussian interval quadrature formula

Summary. We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths. Received July 6, 1999 / Published online August 24, 2000     

On quadrature formulae with maximal trigonometric degree of precision

Summary. In [DD] the problem of existence and uniqueness of a quadrature formula (QF) with maximal trigonometric degree of precision (MTDP) with a fixed number of free nodes and fixed different multiplicities at each node is considered. Even the affirmative answer to the question of existence and uniqueness is useless from a practical point of view if the QF is not explicitly found or if a complete characterization for the nodes and for the coefficients of the QF is not given. On the other hand the problem of the complete constructive characterization of the QF with MTDP is one of the main problems in the theory of numerical integration. In this paper we give a complete constructive characterization for the QF with MTDP in the case of a special type of periodic multiplicities. The results can be considered as a natural generalization of the previous results, which are given in [GO] (one-periodic case of multiplicities) and [DD] (two-periodic case of multiplicities). We evaluate the practical usefulness of the optimal numerical methods, which are obtained. Received June 16, 1995 / Revised version received April 3, 1996     

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Summary. A least-squares mixed finite element method for general second-order non-selfadjoint elliptic problems in two- and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation and the flux approximation consist of piecewise polynomials of degree and respectively. The method is mildly nonconforming on the boundary. The cases and are studied. It is proved that the method is not subject to the LBB-condition. Optimal - and -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented. Received October 15, 1993 / Revised version received August 2, 1994     

Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems

Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented. Received March 11, 1998 / Revised version January 22, 1999/ Published online November 17, 1999     

Numerical integration over a disc. A new Gaussian quadrature formula

Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree . Received August 8, 1996 / Revised version received July 2, 1997     

Error bounds for the large time step Glimm scheme applied to scalar conservation laws

Summary. In this paper we derive an error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2. Received April 10, 2000 / Revised version received January 16, 2001 / Published online September 19, 2001     

The Euler-Maclaurin expansion and finite-part integrals

In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional. Received June 11, 1997 / Revised version received December 15, 1997     

Upper and lower error bounds for plate-bending finite elements

Summary. We compare the robustness of three different low-order mixed methods that have been proposed for plate-bending problems: the so-called MITC, Arnold-Falk and Arnold-Brezzi elements. We show that for free plates, the asymptotic rate of convergence in the presence of quasiuniform meshes approaches the optimal O(h) for MITC elements as the thickness approaches 0, but only approaches for the latter two. We accomplish this by establishing lower bounds for the error in the rotation. The deterioration occurs due to a consistency error associated with the boundary layer – we show how a modification of the elements at the boundary can fix the problem. Finally, we show that the Arnold-Brezzi element requires extra regularity for the convergence of the limiting (discrete Kirchhoff) case, and show that it fails to converge in the presence of point loads. Received June 9, 1998 / Published online December 6, 1999     

Error analysis of the combination technique

Summary. The combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations. In this paper, we present a new technique to analyze the convergence rate of the combination technique. This technique is applied to general second order elliptic differential equations in two dimensions. Furthermore, it is proved that the combination technique for Poisson's equation convergences in arbitrary dimensions. Received September 25, 1997 / Revised version received October 22, 1998 / Published online September 7, 1999     

Relative-error bounds for the LU decomposition via the GTH algorithm

Summary. Recently the author showed that the Grassmann-Taksar-Heyman (GTH) algorithm computes the steady-state distribution of a finite-state Markov chain with low relative error. Here it is shown that the LU decomposition computed in the course of the GTH algorithm also has low relative error. The proof requires a refinement of the methods used in the earlier paper. Received September 2, 1994 / Revised version received July 17, 1995     

Finite volume element methods for non-definite problems

Summary. The error estimates for finite volume element method applied to 2 and 3-D non-definite problems are derived. A simple upwind scheme is proven to be unconditionally stable and first order accurate. Received August 27, 1997 / Revised version received May 12, 1998     

Exact bounds for the uniform approximation by spline interpolants

Summary. This paper completes a result of Reimer (1984) concerning -th-degree cardinal and -periodic interpolation. The method of proof is not restricted to the case of and being odd and seems to be more elementary. Received February 1, 1993 / Revised version received September 14, 1993