Laplace方程分域展开方法

针对电磁场分析中有限元法适应能力强，但计算量大，效率低，而解析法计算量小，但适用范围窄的各自优缺点，将有限元方法与解析方法相结合，提出了一种分域展开的半解析方法用以求解工程电磁场问题。依据龙格定理，把复杂的求解场域分割为若干形状简单的子域，在每个子域内利用本征函数构造半解析展开式逼近方程的解，然后通过子域拼接得到整个场域的解。展开式系数用配点法确定。由于采用解析展开式，逼近效率高，减少了未知数个数；同时由于是局域逼近，具有解函数形式简单，计算量小，矩阵是稀疏的，条件数小，易于求解，实施方便等优点。以二维Laplace问题为研究对象，数值算例验证了方法的有效性，表明该方法花费时间短，计算精确度高。

A subregion expansion method for Laplace's equation

In order to calculate the engineering electromagnetic problems,a new semi-analytical method,sub-region expansion method(SEM) is presented,which takes advantages of both the analytical methods and numerical methods.The basic idea is: firstly,the entire solving domain is divided into simple shaped sub-regions,and in each sub-region a semi-analytical expansion is used to approximate the solution;then all these sub-region expansions are jointed together by the continuity conditions;and finally the coefficients of all expansions are determined by using point-matching technique(PMT).This scheme can overcome some shortcomings of the conventional semi-analytical methods based on the entire-domain-bases,and has a lot of important advantages: the expansion expression is simple and the calculation is reduced;the system matrix is sparse,with a smaller conditioning number,easily to be solved;and lastly the method is easily for implementation.Numerical examples are given to verify the validity of the method.