Parabolic Variational Inequalities: Let where and Suppose: Define:
Parabolic Variational Inequalities (cont’d) Problem 1: Find with such that and for almost all and is such that: for all with a.e. in (0,T).
Parabolic Variational Inequalities (cont’d) Problem 2: Find with such that and for all.
Parabolic Variational Inequalities (cont’d) Equivalence: then Consider a solution of Problem 2, for any and We obtain the solution for Problem 1. Clearly a solution of Problem 1 solves Problem 2.
Parabolic Variational Inequalities (cont’d) a.e. in Theorem Solution to Problem 2 satisfies the linear complementarity system:
Parabolic Variational Inequalities (cont’d) Proof For any non-negative and so, from Problem 2: Which implies that a.e. in
Parabolic Variational Inequalities (cont’d) Hence a.e. in Proof (cont’d) Now let Then for any for sufficiently small so that Conversely, by noting that if satisfies the complementarity system, then, for It is then clear that w solves Problem 2.
Finite Element approximation (FEM) : Space of continuous functions which are linear on each element and which vanish on the boundaries. General Discretisation by FEM for Problem 1: For all (for all interior points) : a piecewise linear basis function. find such that
Finite Element approximation (cont’d) If f is continuous: Otherwise: (for all ) In any case:
Time marching of the discrete system where are nodal vectors. : is a symmetric positive definite matrix which causes the problem (*) to have unique solution.
FEM; Stability Let and let’s assume: Or equivalently that The complementarity problems are equivalent to for all (**) We may take in (**), then
FEM; Stability (cont’d) Stability theorem: Providing the stability condition holds when, there is a constant C, independent of space- and time-steps such that:
FEM; Stability (cont’d) Lemma: is stable under the following conditions The explicit method for which is bounded by
Reference Weak and variational methods for moving boundary problems, C M Elliott & J R Ockendon.