A POSTERIORI ERROR ESTIMATION OF hp-dG FINITE ELEMENT METHODS FOR HIGHLY INDEFINITE HELMHOLTZ PROBLEMS

In this paper, we will consider an hp-finite elements discretization of a highly indefinite Helmholtz problem by some dG-formulation which is based on the ultra-weak variational formulation by Cessenat and Depres. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters h and p. In contrast to the conventional conforming finite element method for indefinite problems, the dG-formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.