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LMS Durham Symposium
Computational methods for wave propagation in direct scattering
Leszek Demkowicz (Texas. USA)
Fully automatic hp-adaptive finite elements for time-harmonic Maxwell's equations
Abstract
1. De Rham diagram for hp Finite Element Spaces
I will begin by reviewing the construction of hp Finite Element (FE) spaces forming the exact sequence and the definition of projection-based interpolation operators that make de Rham diagram commute. I will discuss the optimal p-interpolation estimates in 2D and give a progress report on analogous construction in 3D involving quasi-local interpolation operators.Finally, I will review very briefly, the importance of the diagram in the theory of discretizations of Maxwell's equations.
2. Coding hp Finite Elements - a Programmer's Nightmare
I will guide the audience through the development of an hp- adaptive finite element code in Fortran 90, in the following order:- coding elements of variable order on regular meshes,
- supporting anisotropic h-refinements, constrained approximation,
- supporting multigrid operations,
- automatic hp-adaptivity
3. Automatic hp Adaptivity and 3D Computations
I will discuss various adaptive strategies, focusing on hp-strategy based on minimizing the projection-based interpolation error of a fine mesh solution. The strategy will be illustrated with several 2D involving both H$^1$- and H(curl)-conforming discretizations, and preliminary 3D results. I will present results on coupling goal-oriented adaptivity with automatic hp-adaptivity.Finally, I will present a number of 3D examples obtained in collaboration with Adam Zdunek (Aeronautical Institute of Sweden) and Waldek Rachowicz and Witek Cecot (Technical University of Cracow).