hp-Adaptive Finite Elements for Elliptic And Maxwell Problems Accomplishments and Challenges

Leszek Demkowicz

Institute for Computational Engineering and Sciences (ICES)
The University of Texas at Austin ACES 6.332
Austin, TX 78712
E-mail: leszek@ticam.utexas.edu
ph: (512) 471-4199, fax: (512) 471-8694

I will give an overview on the theory and implementation of hp finite elements with an emphasis on Maxwell equations, and share my experience on the solution of three classes of problems:

  1. Time-harmonic Maxwell equations
  2. Modeling of logging devices.
  3. Computation of waveguide problems,
I will begin with a couple of simple 1D and 2D examples explaining the idea of hp elements and the primary motivation behind using them, including the control of dispersion error, flexibility in handling complex geometries, and the possibility of achieving an unprecedented accuracy through an exponential convergence.

I will discuss then the idea of exact sequences, de Rham diagram and the concept of handling H^1, H(curl), H(div) and L^2 conforming elements in one code.

We will review the main implementation aspects:

  1. data structure combining the advantage of an unstructured initial grid with local h refinements resulting in a locally structured mesh,
  2. support of fully anisotropic refinements necessary for the resolution of vertex and edge singularities and boundary layers,
  3. 1-irregular meshes algorithm and constrained approximation (hanging nodes)
  4. geometrical modeling, initial mesh generation and geometry updates during refinements,
  5. automatic h-, p-, and hp-adaptivity aimed at minimizing interpolation error,
  6. automatic h-, and hp- goal oriented adaptivity,
  7. two grid solver for elliptic and Maxwell problems,
  8. infinite elements.
The three application areas will then be discussed and illustrated with selected numerical examples.