scholarly journals A New Formulation for Solutions of Boundary Value Problems Using Domain-type Approximations and Local Integral Equations

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
V. Sladek ◽  
J. Sladek
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Primary Field ◽  
Type Approximation ◽  
Domain Type ◽  
Local Integral ◽  
New Formulation ◽  
Local Integral Equations

The boundary and domain-type approximations are discussed in boundary integral equation formulations for solution of boundary value problems. A new approach is proposed with using a domain-type approximation of the primary field and collocation of boundary conditions at boundary nodes and local integral representation of the primary field at interior nodes. Two kinds of the domain-type approximation are utilized. The proposed method is illustrated on potential problems in two dimensions.

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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