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An integral equation discontinuous Galerkin method for wide-band and multi-scale problems

Recently, an integral equation discontinuous Galerkin method (IEDG) has been proposed to solve electromagnetic problems. One of the most attractive features of IEDG is the possibility to employ non-conformal mesh for the implementation of the combined field integral equation (CFIE) since the square integrable L2 basis is adopted and no continuity requirement is necessary across boundaries of adjacent elements. Moreover, IEDG provides great flexibility in terms of basis function such that different orders of basis functions with different element types can be mixed with the same discretization to best represent the currents locally. This feature also facilitates the incorporation of adaption techniques to IEDG, either through the use of higher order basis functions (p refinement) or subdivide the elements into multiple smaller pieces (h refinement). All these merits makes IEDG a very attractive candidate for the solution of multi-scale and wide-band problems such that geometry components that are smooth can be discretized using coarser elements while finer discretization can be adopted for more intricate structures. For wide-band simulations, IEDG allows one to adopt the discretization that is generated at a lower frequency for the calculation at higher frequencies through adaptively refining the elements. This helps to reduce the computational burden. However, adoption of higher frequency mesh at lower frequencies could result in the well known ”sub-wavelength breakdown” problem for conventional MLFMM. The computational complexity deteriorates significantly for both multi-scale or low frequency problems. In this paper, IEDG is accelerated by hierarchical multi-level fast multipole method, especially for the problems that involve the dense mesh in a local or global scale. H-MLFMM separates the evanescent physics from radiation physics and employ different transformation of basis accordingly. For the evanescent regime, where densely discretized elements reside, it identifies a reduced set of numerically independent basis functions from the original ones via a skeletonalization algorithm, which in essence is a kernel independent algebraic data spacification algorithm. The reduced set of basis functions are named ”skeletons” or ”skeleton DoFs” and they are adopted to represent the interaction related with the original set of basis functions. Subsequently, significant reduction in memory consumption and CPU time per matrix-vector multiplication (MVP) are achieved through the compressed coupling matrices. With the H-MLFMM incorporated, IEDG offers a very attractive approach to solve for the wide-band EM response from a target via the mesh at midfrequency.