Abstract
In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to deal with the weakly singular kernel of the integral equation, we give the projection method to deal with this part, i.e., using the Lagrange trigonometric polynomials basis to give an approximation of the density function. Although the problems under investigation are well-posed, herein the Tikhonov regularization method is not used to regularize the aforementioned direct problem with noisy data, but to filter out the noise in the corresponding perturbed data. Finally, the effectiveness of the proposed method is demonstrated using a few examples, including a boundary condition with a jump discontinuity and a boundary condition with a corner. Whilst a comparative study with the method of fundamental solutions (MFS) is also given.
Highlights
It is well-known that numerous problems are described by boundary value problems governed by the partial differential equations, which arises in many areas of science, such as wave propagation, vibration, electromagnetic scattering, nondestructive testing, geophysics, and cardiology
The steady state heat conduction equation or the Laplace equation is considered to describe the temperature in heat conducting material
Many problems can be solved by the boundary element method or boundary integral equation method, such as for heat conduction problems [1,2,3,4], acoustic problem [5,6,7,8], elasticity [9], potential problem [10], etc
Summary
It is well-known that numerous problems are described by boundary value problems governed by the partial differential equations, which arises in many areas of science, such as wave propagation, vibration, electromagnetic scattering, nondestructive testing, geophysics, and cardiology. In [23], a modified dual-level fast multipole boundary element method is investigated for large-scale three-dimensional potential problems, and the main idea is based on a dual-level structure to handle the excessive storage requirement and ill-conditioned problems resulting from the fully-populated interpolation matrix of the boundary element method Another effective method, finite element methods, is widely used for the numerical solution, see [24,25]. A novel numerical scheme is proposed to give a stable numerical solution of the Dirichlet problem for the Laplace equation, and it can reduce the condition number of the resulting system. This will display in the last two examples. Some examples are given to show the effectiveness of the method even for a boundary condition with a jump discontinuity
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