Well-posedness of a family of degenerate parabolic mixed equations

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

“…The boundary of Ω is denoted by Γ := ∂Ω and assumed to be Lipschitz continuous. In [3] we have studied the well posedness of the following variational formulation for time-dependent Stokes problem given by Problem 5.1.…”

“…Theorem 5.1. Suppose the assumptions of Theorem 2.1 studied in [3] and let (u, P ) and (u n h , P n h ) N n=1 be the solutions of Problem 5.1 and Problem 5.2, respectively. Furthermore, assume that…”

“…This kind of problems appears in several applications, for instance, to the approximation of the heat equation by means of Raviart-Thomas (RT) elements (see [26]), to the fluid dynamic equations (see [9] and [10]), and to electromagnetic applications (see [1,2,18,28]). Sufficient conditions to the well-posedness of Problem (1.1) with an appropriate initial condition, are given in the recent paper [3]. In that work, the authors combine the linear degenerate parabolic equation theory with the classical Babuška-Brezzi Theory to prove the existence and uniqueness of solution, by assuming some reasonable conditions inspired by the application problems.…”

“…This approximation is obtained by using the finite element method in space with a backward Euler in time. In order to develop the analysis, it is necessary to assume the conditions considered in [3] to ensure the well-posedness of the continuous problem (1.1). Furthermore, to obtain the existence and uniqueness for the fully discrete solutions, the bilinear form induced for the operator A have to satisfy a discrete Gårding-type inequality in the discrete kernel of the bilinear form b and this bilinear form b must satisfy the discrete inf-sup condition.…”

“…The outline of the paper is as follows: In Section 2 we recall the main results given in [3] about the well-posedness of the abstract problem (1.1) and the corresponding analysis for its fully-discrete approximation scheme is presented in Section 3. The results concerning for error estimates of the fully-discrete approximation of the problem are shown in Section 4.…”