Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-Interpolation

Abstract

Abstract. This paper provides a discrete Poincaré inequality in n space dimensions on a simplex K with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on K and all integrals of jumps zero along all interior sides by its Lebesgue norm by C (n) diam(K ). The explicit constant C (n) depends only on the dimension n = 2, 3 in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle ω 0 in the triangulation. The analysis allows the bound of two constants Λ 1 and Λ 3 in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

Nonlinear discontinuous Petrov–Galerkin methods

“…This is already a new result even for the linear cases in [9,13] and opens the door of a convergence analysis of adaptive algorithms via a generalization of [11,14]. This paper contributes the aforementioned equivalent characterizations and a first convergence analysis in the natural norms.…”

“…x h -Bx h p0q{Bt " ξ h that 9 y h exists and is the Riesz representation of´b 1 px h ; ξ h , ‚ q " ap 9 y h , ‚ q in Y h . Therefore, Φpx h ptqq " apy h ptq, y h ptqq{2 is differentiable and the derivative vanishes at t " 0, which leads to 0 " ap 9 y h , y h q for y h -y h p0q.…”

“…Proof The second derivative of Φpx h ptqq reads apB 2 y h {Bt 2 , y h q`}By h {Bt} 2 Y . Recall from the proof of Theorem 2.2 for t " 0, that the Riesz representation 9 y h " By h p0q{Bt satisfies…”

“…It has been discussed in [5,9,13] that the norm of the computed residual }y h } Y " }F´bpv C , q RT ; ‚ q} Yh is almost a computable error estimator for linear problems and this paper extends it to the a posteriori error estimate…”

A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

“…The time reconstructionŴ is well-defined: we have r n +2 unknowns per time interval I n and r n + 1 conditions from (14) and one more condition from (15). It is also unique; we refer to [41, Lemma 2.1] for a proof of the uniqueness, which also shows that the time reconstruction is also globally continuous with respect to the time variable.…”