On Dual-Weighted Residual Error Estimates for p-Dependent Discretizations

This report analyzes the behavior of three variants of the dual-weighted residual (DWR) error estimates applied to the p-dependent discretization that results from the BR2 discretization of a second-order PDE. Three error estimates are assessed using two metrics: local effectivities and global effectivity. A priori error analysis is carried out to study the convergence behavior of the local and global effectivities of the three estimates. Numerical results verify the a priori error analysis. 1 p-Dependence of DG Discretizations Let u ∈ V , where V is some appropriate function space, be the weak solution to a general secondorder PDE described by the semilinear form R(·, ·) : V × V → R. That is, u satisfies R(u, v) = 0, ∀v ∈ V. The space Vh,p is a finite-dimensional space of piecewise polynomial functions of degree at most p on a triangulation Th of domain Ω ⊂ Rn, i.e. Vh,p ≡ {vh,p ∈ L(Ω)| vh,p|K ∈ P (K),∀K ∈ Th}, where P p(K) denotes the space of p-th degree polynomial on element K. A finite element approximation to the problem, uh,p ∈ Vh,p, is induced by the semilinear form Rh,p(·, ·) : Vh,p × Vh,p → R and satisfies Rh,p(uh,p, vh,p) = 0, ∀vh,p ∈ Vh,p. Definition 1.1 (p-Dependence). Let q < p. A semilinear form Rh,p(·, ·) : Vh,p × Vh,p → R is said to be p-independent if Rh,p(wh,q, vh,q) = Rh,q(wh,q, vh,q), ∀wh,q, vh,q ∈ Vh,q ⊂ Vh,p. If a semilinear form is not p-independent, then it is said to be p-dependent. ∗Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 ([email protected], [email protected])