Søren Christiansen

The effective condition number applied to error analysis of certain boundary collocation methods

Abstract The boundary collocation method (BCM) is widely used in the engineering community for the numerical solution of linear boundary value problems. It is commonly stated that the results computed by this method are affected by large rounding errors. This conclusion is based on a simple worst-case analysis using the conventional condition number for linear systems. In this paper we present a refined error analysis for a class of boundary value problems, using the concept of effectively well-conditioned systems. This analysis leads to much smaller a priori bounds for the rounding error in the computed BCM solution. These smaller bounds imply that the numerically computed solution expresses — much more accurately than previously supposed — properties of the ideal BCM solution (computed with infinite precision). Therefore, the intrinsic properties of the BCM can be investigated through numerical computations to a far larger extent than previously supposed, thus allowing for a numerical diagnostic investigation of the BCM. For example, when the BCM is applied to problems with certain geometries, it is observed that the approximation does not converge. Using our new, smaller error bounds, we conclude that the lack of convergence of the BCM solution for these geometries is due to the BCM itself and not to rounding errors.

Numerical solution of an integral equation with a logarithmic kernel

When formulating boundary value problems within different branches of mathematical physics, one encounters an integral equation whose kernel is equal to the logarithm of the distance between two points on a plane, closed, smooth, and simple curve. This equation can be replaced by a system of linear algebraic equations which can be solved numerically.In the present paper we investigate two methods by which this replacement can be performed. Several examples are given in the literature where one of the methods is used. In contrast to this we here put forward a second method, which gives a higher accuracy without requiring more computational effort.

The conditioning of some numerical methods for first kind boundary integral equations

We consider the sensitivity of the matrix equations which arise when solving boundary integral equations of negative order, specifically in case of the typical example of the operator with logarithmic kernel. The sensitivity is expressed using condition numbers and it turns out that the local condition number is a very good indicator, in contrast to the ordinary condition number, traditionally used.

Detecting non-uniqueness of solutions to biharmonic integral equations through SVD

We consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation. The integral operator and also the matrix depend on a real parameter, which may also enter the singular values of the operator and the matrix. When a singular value drops to zero for a certain critical value of the parameter, the corresponding homogeneous integral equation or matrix equation has a nontrivial solution. Based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used. It is therefore possible – with a reasonable accuracy – to detect such critical values for the integral operator simply through a singular-value decomposition of the matrix derived by a quadrature and collocation.

A review of some integral equations for solving the Saint-Venant torsion problem

SummaryIt has been attempted to write down a complete collection of integral equations and functional equations for solving the classical Saint-Venants torsion problem, to a large extent using Greens third identity and a unified notation for the systematic compilation. Some of the equations are well known, others seem to be new. Nearly all the equations are interpreted physically. Known and unknown connections among the equations are pointed out.ZusammenfassungEs wird hier der Versuch gemacht ein vollständiges Verzeichnis von Integralgleichungen und Funktionalgleichungen zur Lösung des klassischen Saint-Venantschen Torsionsproblems aufzustellen-weitgehend mit Hilfe der dritten Greenschen Identität und unter Verwendung einheitlicher Bezeichnungen, um dadurch eine systematische Zusammenstellung zu erreichen. Einige von diesen Gleichungen sind wohlbekannt-andere dagegen werden hier vermutlich zuerst angegeben; beinahe alle Gleichungen werden physikalisch gedeutet. Bekannter und unbekannter Beziehungen zwischen den Gleichungen sind angegeben.

An SVD analysis of linear algebraic equations derived from first kind integral equations

Abstract By means of singular value decomposition (SVD) we investigate the systems of linear algebraic equations which are derived for numerical solution of first kind Fredholm integral equations arising in two-dimensional potential theory. In order to derive a numerical ‘model’ which has the same features and peculiarities as the underlying integral equation, we apply the Galerkin method with orthonormal basis functions. The linear equations are studied with respect to (1) conditioning, (2) accuracy of the computed solution, (3) effective rank, and (4) comparison of nullspaces. As a practical example we then use our numerical method to investigate the equations of a specific geometry for which the analytical solution is not known.

Derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem

We derive and investigate three families of direct boundary integral equations for the solution of the plane, fundamental biharmonic boundary value problem. These three families are fairly general so that they, as special cases, encompass various known and applied equations as demonstrated by giving many references to the literature. We investigate the families by analytical means for a circular boundary curve where the radius is a parameter. We find for all three combinations of equations (i) that the solution of the equations is non-unique for one or more critical radius/radii, and (ii) that this lack of uniqueness can always be removed by combining the integral equations with a suitable combination of one or more supplementary condition(s). We conjecture how the results obtained can, or cannot, be generalized to other boundary curves through the concept logarithmic capacity. A few published general results about uniqueness are compared with our findings.

Modifications of some first kind integral equations with logarithmic kernel to improve numerical conditioning

We consider a pair of integral equations derived by Symm (1967) and Hsiao & MacCamy (1973) for use in various boundary value problems. We investigate the matrix which results when some collocation methods are applied to these equations. We find that thel2-condition number of the matrix can be reduced by performing three minor modifications of the integral equations.ZusammenfassungWir betrachten ein Paar von Integralgleichungen, das von Symm (1967) und Hsiao & MacCamy (1973) zur Verwendung in verschiedenen Randwertaufgaben hergeleitet worden ist. Wir untersuchen die Matrix, die entsteht, wenn bestimmte Kollokationsmethoden auf die Integralgleichungen angewandt werden. Wir finden, daß diel2-Konditionszahl der Matrix verkleinert werden kann, wenn drei einfache Modifikationen der Integralgleichungen durchgeführt werden.

Comparison of four software packages applied to a scattering problem

We investigate characteristic features of four different software packages by applying them to the numerical solution of a non-trivial physical problem in computer simulation, viz., scattering of waves from a sinusoidal boundary. The numerical method used is based on boundary collocation. This leads to highly ill-conditioned linear systems of equations, such that ensuing results may lose significant digits. The packages under consideration, each of which is based on a specific computer arithmetic, are the following: CADNA, PROFIL, MAPLE and MATLAB.

Numerical Treatment of an Integral Equation Originating from a Two-Dimensional Dirichlet Boundary Value Problem

In a previous paper the Kupradze functional equation has been investigated with respect to uniqueness, and it was found necessary to replace the Kupradze equation by a pair of equations. Here we consider the numerical implementation of the pair of equations, and we show how the pair may be treated in order to obtain a satisfactory numerical solution. We also consider the effect of the quadrature error on the solution, whereby we get numerical reasons for a modification of the pair of equations. A major part of the investigation is based on matrix condition numbers formed from the singular values of the matrices.