# DK4: Nonstandard Finite Element Solvers for Second-Order Elliptic Boundary Value Problems

We propose to investigate non-standard finite element schemes for solving second-order elliptic boundary value problems. In the first funding period (2008-2011) we have successfully studied interface-concentrated finite element techniques and finite element methods which are based on boundary element technologies. We will definitely continue the successful research work on the BEM-based FEM. Other non-standard methods for the numerical solution of Partial Differential Equations (PDE) are the so-called Multiharmonic Finite Element Methods (MHFEM), Discontinuous Galerkin (DG) methods and the Isogeometric Analysis (IGA). The MHFEM is a very efficient and highly parallel method for approximating time-periodic (steady-state) solutions of parabolic problems including eddy current problems in electromagnetics. DG-methods allow us to obtain very good approximations to non-smooth solutions arising, e.g., in mathematical imaging and topology optimization. Finally, the IGA that was introduced by T.J.R. Hughes uses the same basis functions (e.g. NURBS) for the representation of the geometry of the computational domain and for the approximation of the solution of the PDEs. The specific choice of the PhD topic connected with one of the above mention non-standard finite element approaches to some specific class of applications depends on the preparatory training, qualifications and knowledge of the PhD candidate. All topics mentioned above require the cooperation with other DK project and with our international research partners.

The construction of fast and highly efficient parallel solvers for the corresponding systems of algebraic equations and their implementation on parallel computers is the ultimate goal of this project. The DK provides a very good infrastructure for supercomputing at the Johannes Kepler University Linz.

Supervisor: Prof. Ulrich Langer