Devising direct regular algorithms of computational potential theory with collocation points inside the solution domain
DOI:
https://doi.org/10.15587/1729-4061.2015.40777Keywords:
regular boundary integral equations, the method of boundary elements/boundary element method, the method of discrete singularities/discrete singularities methodAbstract
Our attempts to systemize and classify the most popular direct collocation algorithms of computational potential theory have resulted in distinguishing two families of algorithms that have not yet been properly researched or used in computational practice. The first family of algorithms is the method of boundary elements with collocation points inside the solution area, while the second one is a direct regular method of discrete singularities with collocation points inside and outside the solution area. Using decomposition of the desired function in the Taylor series along the normal to the border in the vicinity of collocation, we have modified the considered algorithms and obtained significant advantages in comparison with the respective traditional algorithms.
The presented analysis was based on the plane boundary value problems for the Laplace equation. Advantages of the suggested algorithms result from their regular structure that allows to exclude the computation of singular integrals and further program implementation. In addition, the structure permits integration over the real (non-approximated) boundary, which improves the computation accuracy. Eventually, the obtained boundary integral equations are regular integral equations of the second type, which secures their stable solution.
The conclusion is proved by several series of test computation with the use three selected analytically set test functions. The suggested approach can be easily applied to solving broad classes of boundary value problems for differential equations in mathematical physics. The suggested algorithms can be mainly applied in computational mechanics.
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