$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems

Abstract

We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatiallyvarying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.

Robust Adaptive $hp$ Discontinuous Galerkin Finite Element Methods for the Helmholtz Equation

“…For simplicity of the presentation of the equilibrated flux technique, we restrict ourselves to conforming meshes with no hanging nodes. For the necessary modifications to handle irregular meshes we refer the reader to [11].…”

On p-Robust Saturation on Quadrangulations

“…It will prove meaningful to decompose the patch edges E Ta := {e ∈ E T : e ⊂ ω a } as Remark. Our definition of H 1 * (ω a ) differs from its definition in, e.g., [6,8] when a ∈ V ext T . In previous works, functions in H 1 * (ω a ) vanish on the entire part ∂ω a ∩ ∂Ω; in our case, they vanish only on those edges e ⊂ ∂ω a ∩ ∂Ω for which a ∈ e. This altered definition was convenient for our proof, and relevant dual norm properties of the residual in §3 carry over to our case.…”

“…Our goal of adaptive approximation requires us to consider partitions with hanging nodes, which introduce complications. A key contribution in this regard has been made by Dolejší et al in [8].…”

An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

“…[42]. We remark that hp-version a posteriori error indicators, which are based on equilibrated flux reconstruction, may be shown to be robust with respect to the polynomial degree; see, for example, [24,29]. In this latter approach, however, the resulting a posteriori error indicators are implicit in the sense that local problems posed on patches of elements must be numerically approximated in order to compute the elementwise error indicators.…”