In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated.

本文提出了局部Trefftz方法（LTM），以在复杂域中准确有效地解决由Laplace和双调和方程控制的二维边值问题。LTM是通过将经典的间接Trefftz方法和定位方法相结合而形成的，因此LTM不受网格和数值正交的影响，具有解决大规模问题的巨大潜力。对于多重连接域中的问题，由于本地化概念和子域的重叠，与常规的间接Trefftz方法相比，所提出的LTM中的解决方案表达式更简单，更紧凑。在提出的LTM中，既需要内部节点又需要边界节点，并且每个节点处的代数方程表示控制方程或边界条件的满足，可以通过在每个子域上实现Trefftz方法来派生。通过加强每个内部节点上的控制方程和每个边界节点上的边界条件的满足，可以产生线性代数方程的稀疏系统。然后，通过求解稀疏系统，可以有效地获得所提出的LTM的数值解。提供了几个简单连接域和多重连接域中的数值示例，以证明所提出的LTM的准确性和简单性。此外，还同时展示了LTM的极其精确的解决方案。通过求解稀疏系统，可以有效地获得所提出的LTM的数值解。提供了几个简单连接域和多重连接域中的数值示例，以证明所提出的LTM的准确性和简单性。此外，还同时展示了LTM的极其精确的解决方案。通过求解稀疏系统，可以有效地获得所提出的LTM的数值解。提供了几个简单连接域和多重连接域中的数值示例，以证明所提出的LTM的准确性和简单性。此外，还同时展示了LTM的极其精确的解决方案。