Jukka Saranen

Periodic Integral and Pseudodifferential Equations with Numerical Approximation

1 Preliminaries.- 2 Single Layer and Double Layer Potentials.- 3 Solution of Boundary Value Problems by Integral Equations.- 4 Singular Integral Equations.- 5 Boundary Integral Operators in Periodic Sobolev Spaces.- 6 Periodic Integral Equations.- 7 Periodic Pseudodifferential Operators.- 8 Trigonometric Interpolation.- 9 Galerkin Method and Fast Solvers.- 10 Trigonometric Collocation.- 11 Integral Equations on an Open Arc.- 12 Quadrature Methods.- 13 Spline Approximation Methods.

The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane

SummaryIn this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions. It turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed+1. Some numerical examples which conform the theoretical results are presented.

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.

On the collocation method for a nonlinear boundary integral equation

Abstract In this paper we study a potential problem with a nonlinear boundary condition. Using the Green representation formula for a harmonic function we reformulate the nonlinear boundary value problem as a nonlinear boundary integral equation. We shall give a brief discussion of the solvability of the integral equation. The aim of this paper, however, is to analyse the collocation method for finding an approximate solution to this equation. Using the theory of a-proper and a-stable mappings we prove the unique solvability of the collocation equations and the asymptotic error estimates. To do this we assume that the nonlinearity is strongly monotone.

Fast solvers of integral and pseudodifferential equations on closed curves

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order ∥u N - u∥λ ≤ c λ,μ N λ-μ ∥u∥ μ , λ ≤ μ (Sobolev norms of periodic functions) in O(N log N) arithmetical operations.

PARABOLIC BOUNDARY INTEGRAL OPERATORS SYMBOLIC REPRESENTATION AND BASIC PROPERTIES

AbstractIn this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ℝn+1×ℝn+1, and then develop a periodisation procedure for the calculus of symbols on the cylinder

Spline collocation for convolutional parabolic boundary integral equations

\mathbb{T}^n

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

×ℝ. We show Gårdings inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.