Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity

Abstract

In this article we analyse the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the non-conformity of the method and to avoid overpenalisation of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into L 2 (Ω), and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.

Stable Crank–Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity

“…In addition we impose conditions (M1)-(M5) of [1] which are on shape-regularity, boundedness of the polynomial degree, control v h L 4 v h H 1 and the structure of hanging nodes.…”

“…Unlike to [1] which is based on a first-order implicit Euler time-step (leading to a nonlinear system of equations in each time step), here we examine the discretisation in time by a linearised second-order Crank-Nicolson scheme. Crucially, the new, more efficient method inherits stability under low regularity.…”

“…Crucially, the new, more efficient method inherits stability under low regularity. Like in [1], the concentration equation is approximated with a discontinuous Galerkin method, while Darcy's law and the incompressibility condition is formulated as a mixed method. High-order time-stepping for miscible displacement under low regularity has recently also been addressed in [4], however, with a continuous Galerkin discretisation in space and discontinuous Galerkin in time.…”

“…For the data qualification we refer to condition (A1)-(A8) in [1] and for the physical interpretation of the system to [1,2,3]. We point out that D growths proportionally with u:…”

A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem

“…For the Darcy flow, Masud and Hughes [2] introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed formulation. Recently, discontinuous Galerkin for miscible displacement has been investigated by numerical experiments and was reported to exhibit good numerical performance [3,4]. In Hughes-Masud-Wan [5], the method of [2] was extended to the discontinuous Galerkin framework for the Darcy flow.…”