Error analysis of boundary integral methods

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 287-339 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Integral Methods ◽  
Integral Methods

Many of the boundary value problems traditionally cast as partial differential equations can be reformulated as integral equations over the boundary. After an introduction to boundary integral equations, this review describes some of the methods which have been proposed for their approximate solution. It discusses, as simply as possible, some of the techniques used in their error analysis, and points to areas in which the theory is still unsatisfactory.

scholarly journals A hybrid boundary integral/Taylor series approach to some nonlinear equations from fluid mechanics

1987 ◽  
Vol 28 (3) ◽  
pp. 279-286 ◽  
Author(s):  
C. J. Coleman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Boundary Integral Methods ◽  
Integral Methods

AbstractWe show that a combination of Taylor series and boundary integral methods can lead to an effective scheme for solving a class of nonlinear partial differential equations. The method is illustrated through its application to an equation from two dimensional fluid mechanics.

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

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