Discontinuous Galerkin Based Multi-Domain Multi-Solver Technique for Efficient Multiscale Electromagnetic Modeling

Discontinuous Galerkin (DG) methods provide an efficient option for modeling multiscale problems. With the help of the Riemann solver (upwind flux), a discontinuous Galerkin based multi-domain multi-solver technique is introduced in this work for multiscale electromagnetic modeling. Specifically, the proposed technique allows multiple subdomains and multiple solvers. Through multiple subdomains, the original large linear system is reduced into multiple subsystems to solve, thus reducing computational complexity. Different subdomains can be non-conformal with each other. Different element types (tetrahedron, hexahedron, Yee's cell) and element sizes (h-refinement) can be used with different orders of basis functions (p-refinement). For different solvers, the finite element method, spectral element method and finite difference method are incorporated into the proposed technique. As is well known, the finite element method features great mesh flexibility for arbitrarily shaped objects, the spectral element method shows spectral accuracy with high order basis functions, and the finite difference method has great computational efficiency for time domain modeling. With multiple solvers the proposed technique can provide efficient solution for multiscale problems based on different element types. Considering the model geometry, for irregular and complicated structures the finite element method is used with small tetrahedron elements for local refinement, for simple structures the spectral method is used with high order basis functions based on hexahedron elements to exploit its spectral accuracy, and for perfectly matched layer (PML) and layered structures the finite different method is used to improve computational efficiency.

For time domain modeling, firstly hybrid spectral element-finite element method in time domain (hybrid SETD-FETD) is implemented based on the first-order Maxwell's curl equations. To facilitate modeling of electrically small problems, an efficient implicit non-iterative time integration method is proposed based the EB scheme for sequentially ordered systems. Compared with the previous Block-Thomas algorithm, the proposed block Lower-Diagonal-Upper (LDU) decomposition algorithm shows better performance in terms of CPU time and memory, due to the separation of surface unknowns from the volume unknowns.

Then a second-order wave equation based discontinuous Galerkin time domain (DGTD) framework is proposed with a modified Riemann solver to evaluate the flux. Compared with the first-order Maxwell's curl equations based DGTD methods, the new DGTD framework reduces the degrees of freedom for each subdomain by solving the E unknowns plus only surface H unknowns. By contrast, the first-order Maxwell's curl equations based DGTD methods require to solve all the E and B unknowns in each subdomain. To model open problems, a novel coupling method is proposed to incorporate PML into the wave equation based DGTD framework. The PML region is based on the first-order Maxwell's curl equations based DGTD framework with implicit Crank-Nicholson time integration while the physical region follows the wave equation based DGTD framework with implicit Newmark-beta time integration.

To further extend the existing hybrid methods, the hybrid FDTD-SETD-FETD method is proposed by incorporating the finite different time domain method (FDTD). In this work, completely non-conformal mesh is implemented for the first time to hybridize FDTD, SETD and FETD for 3D modeling. Based on the DGTD framework, a buffer zone is introduced between FDTD and SETD/FETD to facilitate the coupling procedure. A global leapfrog time integration is implemented to validate the proposed hybrid FDTD-SETD-FETD method, and the implicit-explicit time integration is proposed to improve its performance for practical applications. To remove the buffer zone, a more advanced hybridization technique is introduced, which shows better performance in terms of CPU time and memory. The corresponding explicit leapfrog and implicit-explicit time integration scheme are also given for the new hybrid FDTD-SETD-FETD method without buffer.

For frequency domain modeling, based on the second-order wave equation and time harmonic assumption, a discontinuous Galerkin frequency domain (DGFD) method is introduced with the Riemann solver for anisotropic media. To improve the accuracy and efficiency, a mixed total field/scattered field DGFD (TF/SF DGFD) formulation is given. For subdomains with sources and receivers, the scattered field based DGFD is used to improve accuracy while for the remaining subdomains the total field DGFD is used to improve efficiency. With TF/SF DGFD, the scattered field at the receivers can be directly obtained. In addition, some useful boundary conditions, including scattering boundary condition (SCBC), surface impedance boundary condition (SIBC) and impedance transition boundary condition (ITBC), are incorporated into the proposed DGFD framework to further improve its performance for geophysical exploration problems. SCBC is used to truncate the physical region, SIBC is for approximating the effect of thick imperfect conductor, and ITBC is for approximating thin imperfect conductor as a surface. ITBC is used for the first time in this work for fracture modeling, and shows good agreement with the reference.

Finally, a domain decomposition based inversion method is proposed based on the DGFD forward solver for inverse scattering problems. The inversion algorithm is given based on the Gauss-Newton method, and more importantly, the formulation related to domain decomposition is derived. With the advantage of domain decomposition, an adaptive inversion procedure is introduced to gradually improve the inversion resolution and accuracy.