Boundary Integral Equations
Boundary Integral Equations
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DOI：
10.1007/0-387-34042-4_1
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
G. Hsiao;W. Wendland
中科院分区：
文献类型：
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作者：
G. Hsiao;W. Wendland
当基本偏微分方程解已知时，二阶偏微分方程解可以用一定的表面势和体积势来描述。尽管对于一大类偏微分算子，这样的基本解的存在是可以保证的，例如，见[53]，但在一般情况下，基本解的显式构造是一项更困难的任务。因此，我们在这里只考虑常系数偏微分算子。特别是，我们仅限于Laplace算子、Helmholtz算子、线性弹性静力学和Stokes系统，其中包括边界积分方程组和边界元方法的最重要的应用。当使用源于格林第二公式的表示公式时，或者当考虑间接表面势法时，必须从给定的边界条件中寻找未知密度函数。这是通过将相应的迹运算符应用于表面势和体积势来实现的，从而产生要求解的适当的边界积分方程组。根据给定的边界条件，可以推导出第一类或第二类边界积分方程组的不同形式。虽然在连续水平上，所有的边界积分方程都等价于原边值问题，因此，在应用数值格式获得近似解时，它们具有完全不同的性质。在本章中，我们概述了利用边界积分方程解二阶边值问题的直接和间接重构，并讨论了所涉及的所有边界积分算子的映射性质。由此，我们可以推导出所得到的边界积分方程解的唯一性以及解对给定边界数据的连续依赖性。
The solutions of second order partial differential equations can be described by certain surface and volume potentials when a fundamental solution of the underlying partial differential equation is known. Although the existence of such a fundamental solution can be guaranteed for a wide class of partial differential operators, see for example [53], the explicite construction of fundamental solutions is a more difficult task in the general case. Hence, we consider here partial differential operators with constant coefficients only. In particular, we restrict ourselves to the Laplace operator, the Helmholtz operator, and the systems of linear elastostatics and of Stokes, which include the most important applications of boundary integral equations and boundary element methods. When using either a representation formula stemming from Green’s second formula or when considering indirect surface potential methods, one has to find unknown density functions from the given boundary conditions. This is done by applying the corresponding trace operators to the surface and volume potentials yielding appropriate boundary integral equations to be solved. Depending on the given boundary conditions one can derive different formulations of first or second kind boundary integral equations. Although on the continuous level all boundary integral equations are equivalent to the original boundary value problem, and, therefore, to each other, they admit quite different properties when applying a numerical scheme to obtain an approximate solution.In this chapter we give an overview of direct and indirect reformulations of second order boundary value problems by using boundary integral equations and discuss the mapping properties of all boundary integral operators involved. From this we can deduce the unique solvability of the resulting boundary integral equations and the continuous dependence of the solution on the given boundary data.