有限差分法解电磁场方程的共轭梯度法三角阵预处理器

通过考察电磁场微分方程经非均匀网格离散后的有限差分方程组矩阵,建立了函数偏微分运算与离散向量矩阵相乘运算的对应关系,给出了差分方程组矩阵对应于微分算子的分解式,并据此提出了共轭梯度法的三角阵预处理器。此外,还提出了对不同的边界条件、求解域内部边界、介质分界面和时谐场方程的处理技术以便应用该预处理器。数值计算结果验证了本文算法的正确性,展示了其十分明显的加速收敛效果,表明了本文算法有线性的存储复杂度和几乎线性的计算复杂度,可有较广泛的应用。本文中将算子细节和矩阵细节对应的基本思想对构造其它高效预处理器具有借鉴作用。

The Triangular Matrix Preconditioners of the Conjugate Gradient Method for Solving Equations of Electromagnetic Fields Using Finite Difference Method

In this paper,correspondence between partial differential operations to a function and matrix multiplications to the discrete vector is established by exploring the matrix of finite difference system obtained from the nonuniform discretization of a differential equation of electromagnetic fields.A decomposition representation of the matrix is derived,based on which,the triangular matrix preconditioner for the conjugate gradient method is presented.Besides,some techniques for boundary conditions,interior boundaries,interfaces between different dielectric media and time harmonic equation are proposed for the employment of this preconditioner.The correctness are verified by the numerical results which show the obvious effect of convergence acceleration,the linear storage complexity and the almost linear computational complexity of the preconditioned algorithm.It can be concluded that the algorithm can be widely used.The basic idea to correspond operator details with matrix details presented in this paper is worthy of being considered for the constructions of other effective preconditioners.