计算数学 1980, 2(1) 100-105 DOI: ISSN: 0254-7791 CN: 11-2125/O1 | |||||||||||||||||||||||||||||||||||||||
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论文 |
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论微分与积分方程以及有限与无限元 | |||||||||||||||||||||||||||||||||||||||
冯康, | |||||||||||||||||||||||||||||||||||||||
中国科学院计算中心, | |||||||||||||||||||||||||||||||||||||||
摘要：
1 椭圆微分方程的边界值问题可以有种种不同的数学成型,在理论上等价,但在实践上不等效。有限元方法成功的一个关键就是合理选取了变分的数学型式。举例来说,取调和方程的第二类边界问题,定义于区域Ω,具有光滑边界Г: | |||||||||||||||||||||||||||||||||||||||
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DIFFERENTIAL VERSUS INTEGRAL EQUATIONS AND FINITE VERSUS INFINITE ELEMENTS | |||||||||||||||||||||||||||||||||||||||
Feng Kang (Computing Center,Academia Sinica) | |||||||||||||||||||||||||||||||||||||||
Abstract:
Boundary-value problems of elliptic equations may have many different mathematical formulations, equivalent in principle but not equally efficient in practice. For example, Neumann problem of Laplace equations (1),(2) is equivalent to the variational problem (3),(4). The judicious use of the latter formulation leads to the success of the FEM. The problem can also be formulated in terms of integral equations, even in many ways. They have generally the advantage of the reduction both of dimension by 1 and of the infinite domain to the finite, at the expense of increased analytical difficulty. The most wall-known reduction is the Fredholm integral equation of the second kind(5), for which w is to be solved and gives the original solution through the integral formula(6). The corresponding integral operator maps H~s(Ω)→H~s(Ω) and is, in general, not self-adjoint, so one of the characteristic and useful properties of the original problem is lost. A less-known reduction to integral equation is in the form(7), for which the boundary value u_0 of the solution u to the orginal problem is to be solved and gives u through the integral formula(8). The kernel K has the advantage of being self-adjoint and is derived from the Green’s function by double differentiation so is highly singular. It is of the type of the finite part of the divergent integral in the sense of Hadamard and maps H~s onto H~(s-1)and is thus desmoothing by orde r 1. This is advantageous rather than defective to the solution stability. Furthermore, the variational formulation equivalent to(7) is(11), (12)which can be obtained from (3),(4) through elimination of interior values of u by means of Green’s function. This form of reduction to integral equation is related to the original problem in a more natural and direct way, so it will be regarded as canonical and is more desirable in numerical approach. In fact, the idea of canonical reduction is implicitly used in FEM practice as technique of substructures. The elimination of the internal degrees of freedom is precisely a discrete analog of the canonical reduction and the resulted algebraic system containing solely the boundary degrees of freedom is precisely a discrete analog of the Hadamard integral equation. Recently, an elegant scheme of infinite similar elements has been proposed for the solution of crack singularity problems. They are equally well suited for concave corners, intersection of several interfaces, infinite domains and also the usual closed domain of regularity. For all these cases, it can be shown that, under certain uniformity condition, a conforming finite element in infinite similar triangulation converges with its nominal order of accuracy without deterioration. This elimination of infinite number of the interior degrees of freedom is another example of discrete analog of the canonical reduction. Fig. 1 affords an example problem containing various kinds of singularity and infinite domain. It can be grossly divided into 5 substructures using infinite triangulation for each. This suggests an economy of problem preparation, storage space and volume of computation. Fig. 2 is an infinite triangulation of the unit circle, the finite algebraic system for the boundary unknowns after the elimination of infinite many interior unknowns gives a discrete analog of the Hadamard integral equation(14) with the finite part kernel 1/sin~2θfor the unit circle. This is an example of solving integral equation without explicit use of integral equation, also that of treating finite parts without explicit presentation of finite parts. | |||||||||||||||||||||||||||||||||||||||
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收稿日期 修回日期 网络版发布日期 1980-01-14 00:00:00.0 | |||||||||||||||||||||||||||||||||||||||
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