Tamtam Proceedings - lamsin

A max-plus finite element method 73for 1 ≤ i ≤ p and 1 ≤ j ≤ q. Here, argmax{z j + w i } denotes the set of x such thatz j (x) + w i (x) = 〈z j | w i 〉.5. Comparison with the method of Fleming and McEneaneyFleming and McEneaney proposed a max-plus based method [5], which also uses aspace W h generated by finite elements, w 1 , . . . , w p , together with the linear formulation(5). Their method approaches the value function at time t, v t , by W h µ t , where W his as above, and µ t is defined inductively byµ 0 = W h \φ and µ t+δ = ( W h \(S δ W h ) ) µ t ,for t = 0, δ, . . . , T − δ. This can be compared with the limit case of our finite elementmethod, in which the space of test functions Z h generates the set of all functions. Thislimit case corresponds to replacing Zh ∗ by the identity operator in (10), so thatWe prove that W h µ t ≤ W h λ t ≤ v t , for t = 0, δ, . . . , T .λ t+δ = W h \(S δ W h λ t ) . (14)6. Error AnalysisThe error of the finite element method is controlled by the projection errors, ‖P Wh v t −v t ‖ ∞ and ‖P op−Z hv t − v t ‖ ∞ , and by the approximation errors, ‖[S δ w i ] ∼ − S δ w i ‖ ∞ ,and (B ∼∼h) ji − (B ∼ h ) ji. The following theorem gives an error estimate of the effectivemax-plus finite element method implemented with the approximation B ∼∼hof B h , for asubclass of problems in dimension 1.Theorem 1. Let X = [−b, b] ⊂ R. Assume that there exist L > 0 and c > 0 such thatthe value function at time t, v t , is L-Lipschitz continuous and 1 c-semiconvex for all t > 0.Let us choose the finite elements w i , such that w i (x) = − 1 2c (x − ˆx i) 2 , where ˆx i are thepoints of the regular grid (Z∆x) ∩ [−(b + cL), (b + cL)]. Let us choose the test functionsz j , such that z j (x) = −A|x − ˆx j |, A ≥ L, where ˆx j are the points of the regular grid(Z∆x) ∩ [−b, b]. For t = 0, δ, . . . , T , let vh t be the approximation of vt given by theeffective max-plus finite element method, implemented with the approximation B ∼∼hofB h . Then, under regularity assumptions there exists a constant K > 0 such that, for δsmall enough, ‖vh T − vT ‖ ∞ ≤ K( √ δ + ∆xδ ).This error estimate is of the same order as in the case of existing discretization methods,see [4].Example 2. We consider the case where T = 1, φ ≡ 0, X = [−1, 1], U = [−1, 1],l(x, u) = −3(1 − |x|) and f(x, u) = u(1 − |x|). We represent below the solution givenby our algorithm in the case where δ = 0.05, ∆x = 0.02, A = 2 and c = 1.1.TAMTAM –Tunis– 2005