Efficient large scale electromagnetic simulations using dynamically adapted meshes with the discontinuous Galerkin method

A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials (p-adaptation) as well as their combination. The computation of the approximation within locally adapted elements is based on projections between finite element spaces (FES), which are shown to preserve an upper limit of the electromagnetic energy. The formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency. Error and smoothness estimates based on interface jumps are presented and applied to the fully hp-adaptive simulation of two examples in one-dimensional space. A full wave simulation of electromagnetic scattering from a radar reflector demonstrates the applicability to large scale problems in three-dimensional space.

[1]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[2]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[3]  Thomas P. Wihler,et al.  An hp-adaptive strategy based on continuous Sobolev embeddings , 2011, J. Comput. Appl. Math..

[4]  Endre Süli,et al.  hp-Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems , 2001, SIAM J. Sci. Comput..

[5]  Thomas Weiland,et al.  A hybrid Finite Integration-Finite Volume Scheme , 2010, J. Comput. Phys..

[6]  Gary Cohen,et al.  A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time domain , 2006, J. Comput. Phys..

[7]  E. Süli,et al.  A note on the design of hp-adaptive finite element methods for elliptic partial differential equations , 2005 .

[8]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[9]  Thomas Weiland,et al.  Discontinuous Galerkin methods with transient hp-adaptation , 2010, 2010 URSI International Symposium on Electromagnetic Theory.

[10]  L. Demkowicz.  One and two dimensional elliptic and Maxwell problems , 2006 .

[11]  L. Fezoui,et al.  Discontinuous Galerkin time‐domain solution of Maxwell's equations on locally‐refined nonconforming Cartesian grids , 2005 .

[12]  J. Tinsley Oden,et al.  A parallel hp -adaptive discontinuous Galerkin method for hyperbolic conservation laws , 1996 .

[13]  Bernardo Cockburn.  Discontinuous Galerkin methods , 2003 .

[14]  Bernardo Cockburn.  A Simple Introduction to Error Estimation for Nonlinear Hyperbolic Conservation Laws , 1999 .

[15]  L. Fezoui,et al.  Convergence and stability of a discontinuous galerkin time-domain method for the 3D heterogeneous maxwell equations on unstructured meshes , 2005 .

[16]  Ilaria Perugia,et al.  The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations , 2003, Math. Comput..

[17]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[18]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[19]  Jörg Stiller,et al.  Stabilized High-Order Discontinuous Galerkin Methods for Aeroacoustic Investigations , 2009 .

[20]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[21]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[22]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[23]  D. Schötzau,et al.  Stabilized interior penalty methods for the time-harmonic Maxwell equations , 2002 .

[24]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[25]  Ivo Dolezel,et al.  Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM , 2008, Math. Comput. Simul..

[26]  J. Remacle,et al.  Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws , 2004 .

[27]  B. Bergk,et al.  Magnetic-field- and temperature-dependent Fermi surface of CeBiPt , 2006 .

[28]  Seizo Tanaka,et al.  A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals , 2010 .

[29]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[30]  J. Hesthaven,et al.  Nodal high-order methods on unstructured grids , 2002 .

[31]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[32]  Rainald Löhner,et al.  On the Computation of Steady-State Compressible Flows Using a DG Method , 2009 .

[33]  Ilaria Perugia,et al.  Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator , 2005 .

[34]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[35]  Ivo Babuska,et al.  The p and h-p Versions of the Finite Element Method, Basic Principles and Properties , 1994, SIAM Rev..

[36]  I. Doležel,et al.  Adaptive hp-FEM with Arbitrary-Level Hanging Nodes for Maxwell's Equations , 2010 .

[37]  Thomas Weiland,et al.  Accurate modelling of charged particle beams in linear accelerators , 2006 .

[38]  Antonio Huerta,et al.  One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods , 2013 .

[39]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[40]  J. S. Hesthaven,et al.  Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method , 2011, 1102.3190.

[41]  Leszek Demkowicz,et al.  Toward a universal adaptive finite element strategy part 3. design of meshes , 1989 .