Discontinuous Immersed Finite Element Methods for Interface Problems

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

Immersed finite element (IFE) methods have been developed to solve problems with discontinuous coefficients and they are very competitive because of two essential features: (1) they allow structured meshes and avoid mesh distortion associated with Lagrangian and interface-tracking methods; (2) their basis functions incorporate the interface jump conditions imposed by physics. The objectives of this project are to combine IFE methods with the versatile discontinuous Galerkin (DG) finite element methods to efficiently and accurately capture solution discontinuities, simplify h and h-p adaptivity, and yield efficient parallel methods for interface problems arising from modeling multi-scale and multi-physics procedures in engineering and sciences. The PIs of this project plan to construct and analyze DG methods with IFE spaces that attain optimal or near optimal convergence rates under h and h-p refinement on meshes not necessarily aligned with interfaces. The PIs plan to focus on three representative types of boundary value problems with discontinuous coefficients in two and three dimensions: (i) Second-order elliptic problems, (ii) Linear elasticity systems and fourth-order equations for beams and plates, (iii) Maxwell's equations. The PIs will investigate problems with linear and curved interfaces on meshes consisting of triangles and quadrilaterals in two dimensions, tetrahedrons and hexahedrons in three dimensions. The PIs will perform error analysis and investigate the convergence of the IFE solution to the true solution under h, p and h-p refinement. The PIs will also investigate superconvergence properties of discontinuous IFE solutions and use these results to construct efficient and reliable a-posteriori IFE error estimators for assessing solution quality and guiding adaptivity. Simulating a multi-scale/multi-physics phenomenon often involves a domain consisting of different materials and leads to an interface problem consisting of partial differential equations with discontinuous coefficients, boundary (and initial) conditions, and jump conditions required by pertinent physics across the material interfaces. It is well known that efficiently solving interface problems is critical for numerical simulations in many applications of engineering and sciences, including flow problems, electromagnetic problems, shape/topology optimization problems, to name just a few. IFE methods are competitive methods for solving interface problems and the methods developed in the proposed research projects will have direct impacts on numerical simulations in electric propulsion of plasma engine design/research, optimal packaging of electronic devices, efficient and better image reconstruction in computer tomography, polymer matrix carbon fiber reinforced composites and related material technologies in aerospace and space structures, non-destructive/non-invasive detection of suspicious materials in security check, design of optimal shapes for lighter and stronger structures, and many other application areas of great federal interests.