Simplifying numerical solution of constrained PDE systems through involutive completion

Abstract

Abstract. When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.Mathematics Subject Classification. 35G15, 35N10, 65M60, 65N30.

Involutive formulation and simulation for electroneutral microfluids

“…However, for our purposes it is sufficient to explain the idea in concrete terms and indicate how to work with this notion constructively. For more details we refer to [5,8,9,12,13,17,19,21].…”

“…The drawback is that we have introduced an extra variableỹ which increases the computational cost. However, we can useỹ in error control as explained in [12].…”

“…Because we have already analysed the augmented Stokes system in our previous papers [12,13] we will only discuss below how our formulation improves the quality of the solutions of concentrations of species c i . This is also justified by the fact that in this model the main interest and the main computational difficulty is related to concentrations while the motion of the ambient fluid is relatively unimportant and in any case considerably easier to compute.…”

“…Because the geometry is simple we used quadrilateral meshes. First we have Stokes or augmented Stokes solver for u and p. The classical Stokes solver needs no comment and the augmented Stokes system is described in [12,13]. Note that we have used equal order discretizations for u and p in the augmented system because in our approach we do not need to worry about infsup-condition.…”

“…Our method, already used in [12,13], uses the involutive and its associated augmented form of the system in numerical computations. The augmented form can be constructed once the involutive system and its compatibility operator are known.…”