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Breaking Spaces and Forms for the DPG Method and Applications Including Maxwell Equations
Computers & Mathematics with Applications
  • Carsten Carstensen, Humboldt-Universität zu Berlin
  • Leszek Demkowicz, University of Texas at Austin
  • Jay Gopalakrishnan, Portland State University
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  • Harmonic functions,
  • Finite element method,
  • Numerical analysis,
  • Maxwell equations,
  • Electromagnetic fields
Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and a spectrum of forms in between.

This article was accepted for publication by Elsevier. The definitive version can be found here.

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Citation Information
Carstensen, C., Demkowicz, L., & Gopalakrishnan, J. (2016). Breaking spaces and forms for the DPG method and applications including Maxwell equations. Computers & Mathematics with Applications, 72(3), 494–522.