A posteriori error estimates of mixed finite element solutions for fourth order parabolic control problems

Abstract

In this paper, a fourth order quadratic parabolic optimal control problem is analyzed. The state and co-state are discretized by the order k Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k (k ≥ 0). At last, the results of a posteriori error estimates are given in Lemma 2.1 by using mixed elliptic reconstruction methods.

Weak Galerkin finite element methods for a fourth order parabolic equation

“…In Table 1, the suboptimal-like convergence order of WG with V h (2, 2, 1) is shown in the |||⋅||| and L 2 -norm, which is O(h) and O(h 2 ), respectively. While Table 2 presents the optimal convergence order with V h (3,3,2). Thereinto, the order of the L 2 -norm is O(h 4 ) and the one of |||⋅||| is O(h 2 ).…”

“…This problem could be solved in two ways: one is to use mixed finite element methods, and the other is nonconforming finite element methods. The key point of mixed finite element methods is to rewrite the fourth order parabolic equation as a coupled system of two second‐order partial differential equations, which can be solved using the standard Galerkin finite element method . However, the mixed finite element method needs the solution of a saddle point problem for which a certain stability condition should be satisfied in the algorithmic development and analysis.…”

“…Let u ∈ H max{k+1,4} (Ω) be the true solution of (1.1)-(1.3), u h ∈ V h be the numerical solution of (3.6) with finite element functions of order k ≥ 2, and = min(k,3). Then there exists a constant C such that…”