Coulomb 规范下地电磁场的自适应有限元模拟的理论分析

地电磁场的直接求解法存在伪解现象，且电磁场分量在界面上的不连续性与节点型有限元的基本要求矛盾. 本文将Coulomb 规范下磁矢量势-电标量势与自适应有限元相结合，提出了地球物理电磁场计算的快速、高精度方法. 首先从地电磁场一般边值问题出发，给出了Coulomb 规范下磁矢量势-电标量势的公式系统，分析了求解域内势的连续性. 采用Galerkin 加权余值法推导出积分弱解形式和Delaunay非结构化四面体单元时Hierarchal 基函数的有限元方程. 基于超收敛恢复技术，提出了适用于电磁场的后验误差估计方法，阐述了地电磁场自适应计算的策略及迭代算法，分析了计算时间消耗和误差收敛性质，表明本文方法可以用最优的计算资源得到呈拟指数收敛到准确解的数值结果，从而为后续的数值计算奠定了理论基础.

Theoretical analysis of geo-electromagnetic modeling on Coulomb gauged potentials by adaptive finite element method

There are several serious problems in geo-electromagnetic modeling by using direct solving method at present.The first problem is that it often gives a psedo-solution that does not adequately describe the existing geo-electromagnetic field.The second problem is that the discontinuity of the electromagnetic field at the conductivity interface limits the application of the node-based finite element method.In this paper we propose a new,fast and accurate method of forward modeling using the Coulomb gauged magnetic vector potential and electric scalar potential concept and the newly developed adaptive finite element method.Based on the typical electromagnetic boundary value problem,we introduce a formulation system for magnetic vector potential and electric scalar potential under Coulomb gauged potential,and have analyzed the continuity properties of the vector and scalar potentials.Adopted the Galerkin weighted residual technique,we have formulated the weak integral form of magnetic vector and scalar potentials,and then developed the finite-element linear equations based on the unstructured Hierarchal tetrahedron finite element concept.By using the super-convergence recovery techniques,we are able to estimate the element and global errors on current mesh.We have also done some analyses on this adaptive finite-element method and found that we could achieve a more accurate numerical solution with an optimal computing cost.This new technique presented in this paper can be used as a theoretical guide in a wide range of geophysical electromagnetic explorations.