Spectral element method based on domain decomposition for three-dimensional DC resistivity modeling

We develop a new spectral-element method by incorporating a domain-decomposition technique, referred to as the dual-primal spectral-element tearing and interconnecting method (SETI-DP), for high-precision forward simulation of the 3D direct current (DC) resistivity method. To handle the terrain and complex underground structures, we use unstructured tetrahedral meshes to divide the computational domain. These meshes are then partitioned into nonoverlapping subdomains. By applying a continuity condition of the current density to be solved, all subdomains become coupled to each other. Using a Lagrange multiplier, the global equation system for our forward modeling can be simplified into that for the subdomain interfaces, and the continuity conditions on the subdomain interfaces are obtained by solving these subdomain interface equations. Finally, the forward problem for each subdomain is solved independently using these boundary conditions. Numerical experiments indicate that our algorithm can greatly improve computational efficiency and reduce memory consumption without losing accuracy when choosing the appropriate number of subdomains. We further apply our method to the forward simulation of a tunnel detection problem. The numerical results demonstrate that our new SETI-DP method can effectively simulate the tunnel’s complex structures, and the DC resistivity method can detect the adverse geologic bodies in front of the tunnel face.