An a posteriori error estimator and hp-adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains

This paper presents an a posteriori error estimator and hp-adaptive strategy for the Helmholtz equation in exterior domains. These technologies are integral components of solution adaptive finite element analyses, with application to, e.g., time-harmonic exterior acoustics problems. The error estimator is an explicit function of residuals (no local problems need to be solved), and is in the form of an upper bound on the global L(2)-norm of the error. A posteriori error bounds are derived for two finite element discretizations: the Galerkin and Galerkin least-squares (GLS) formulation. The hp-adaptive strategy converts the estimated error distribution, which is extracted from the global error norm, into a distribution of element size h and polynomial order p for the subsequent adaptive mesh. The principle of equidistribution of element error is used to guide the {h, p} distribution, leading to an efficient adaptive mesh. The adaptive strategy is a simple algorithm which computes pointwise values of h and p, and allows for simultaneous refinement of both. In addition, the adaptive strategy does not require knowledge of the global scaling constant appearing in the error estimator, which greatly simplifies the computation.