Abstract
Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.
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This work was carried out in the framework of the DFG under grant no. HI 1466/7-1 “Free Boundary Problems and Level Set Methods”, the DFG-funded SFB-TRR 154 Subproject B02, as well as the Research Center Matheon supported by the Einstein Foundation Berlin within projects OT1, SE5, and SE15/19. The authors further gratefully acknowledge the support of the DFG through the DFG-SPP 1962: Priority Programme Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization within projects 10, 11 and 13.
Appendices
Appendix A: Lower Solutions for VIs
The following result is due (to the best of our knowledge) to Bensoussan and is included for the sake of completeness.
Proposition 5
Let\(A:{H_{0}^{1}}({\Omega })\to H^{-1}({\Omega })\)belinear, bounded and uniformly monotone. Additionally, suppose thatif\(v\in {H_{0}^{1}}({\Omega })\),then 〈Av−,v+〉≤ 0.Letφi ∈ L∞(Ω) withi = 1, 2 be such 0 ≤ φ1 ≤ φ2a.e.,
and suppose that fi ∈ L2(Ω) with i = 1, 2 and f1 ≤ f2a.e.. Then, y1 ≤ y2a.e., where\(y_{i}=\mathbb {S}(A, f_{i}, \mathbf {K}(\varphi _{i}))\).
Further, let\(\varphi \in L_{+}^{\infty }({\Omega })\)and f ∈ L2(Ω). If z ∈K(φ) satisfies
we say z is a lower solution for the triple (A, f, K(φ)). For any lower solution z, we have that\(z\leq \mathbb {S}(A, f, \mathbf {K}(\varphi ))\).
Proof
Since yi ∈K(φi) for i = 1, 2, then 0 ≤ v1 := min(y1,y2) = y1 − (y1 − y2)+ ≤ ψ1 and 0 ≤ v2 := max(y1,y2) = y2 + (y1 − y2)+ ≤ ψ2. Hence, from 〈Ayi − fi,vi − yi〉≥ 0 for i = 1, 2, we obtain
Subtracting the second inequality from the first one, we observe
since f1 − f2 ≤ 0. Since A is uniformly monotone and 〈Av−,v+〉≤ 0, for all \(v\in {H_{0}^{1}}({\Omega })\), we obtain the following chain of inequalities:
Therefore, (y1 − y2)+ = 0 a.e., that is, y1 ≤ y2 a.e. in Ω.
Let \(y= \mathbb {S}(A, f, \varphi )\), so y ∈K(φ) and
Replacing v = y − ϕ with \( \phi \in {H_{0}^{1}}({\Omega })\) and ϕ ≥ 0 a.e. in Ω, we observe that \(y= \mathbb {S}(A, f, \mathbf {K}(\varphi ))\) is a lower solution for the triple (A,f,K(φ)). Now we prove that if z is an arbitrary lower solution, the z ≤ y a.e. in Ω. Let ϕ = (z − y)+ and v = max(y,z) = y + (z − y)+ on (64) and (66), respectively, then
These are exactly the same inequalities as in (65). Therefore, we have that (z − y)+ = 0, i.e., z ≤ y a.e. in Ω. □
Appendix B: Proof of Lemma 1
Proof
Consider first a and i = 1. Let \(w\in \mathscr{K}({\Psi })\) and note that the condition “\(w^{N}(t)\in \mathbf {K}({\Phi }(t_{n-1}^{N},u_{n-1}^{N}))\) with \(t\in [t_{n-1}^{N},{t_{n}^{N}})\)” is equivalent to
Denote by \(\{\tilde {u}^{N}\}\) the convergent subsequence obtained in Theorem 2, i.e., \(\tilde {u}^{N}\rightarrow u^{*}\) in C([0,T]; L2(Ω)). Then, by the inequality in (19) we also have that
By Assumption 1, we have that Φ : [0,T] × L2(Ω) → L∞(Ω) is uniformly continuous, i.e., for any 𝜖 > 0, there exists δ(𝜖) > 0 such that
Therefore, for sufficiently large N we have that
for all τ ∈ [0,T]. Recall that by assumption we have that the mapping Φ satisfies: 1) Φ(t,v) ≥ ν > 0 a.e. in Ω, for a.e. t ∈ [0,T] and all v ∈ L2(Ω). 2) It is non-decreasing in both variables. 3) T↦Φ(T,v) maps bounded sets in L2(Ω) into bounded sets in L∞(Ω). Then, we define \(\varphi _{N}(t,x):=\hat {{\Phi }}(t, u_{-}^{N}(t))(x)\) and φ(t,x) := Φ(t,u∗(t))(x) with (t,x) ∈ Q := [0,T] ×Ω. It follows that φN,φ ∈ L∞(Q) and also
Now, we prove that for any η ∈ (0, 1), there is an N(η) such that
for N ≥ N(η) . In fact, let η ∈ (0, 1) be arbitrary, and consider the sets
Then, for almost all z ∈ QN, we have
But since \(|\varphi - \varphi _{N}|_{L^{\infty }(Q)}\rightarrow 0\), there exists \(N(\eta )\in \mathbb {N}\), such that |QN| = 0 for all N ≥ N(η).
Let {ηj} be a monotonically increasing sequence in (0,1) such that \(\lim _{j\rightarrow \infty }\eta _{j}= 1\). Let \(w\in L^{2}(0, T; {H_{0}^{1}}({\Omega }))\) satisfy |∇w(t)(x)|≤ φ(t,x). Then, wj := ηjw fulfils
for almost all (t,x) ∈ Q. Finally, \(|w^{j}-w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}=(1-\eta _{j})|w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}\leq (1-\eta _{j})|\varphi |_{L^{\infty }(Q)} \rightarrow 0\) as j →∞. This proves the statement concerning \(w\in \mathscr{K}({\Psi })\).
Next, we focus on a and i = 2. For the same sequence {ηj} as before, suppose \(w\in L^{2}(0, T; {H_{0}^{1}}({\Omega }))\) is arbitrary and such that \(w\in \mathscr{K}^{\pm }({\Psi })\). Then wj(t) = ηjw(t) belongs to \(\mathbf {K}^{\pm }(\varphi _{N(\eta _{j})}(t,\cdot ))\), i.e., for d(x) := dist(x,∂Ω)
(where we have omitted “(t, x)” for the sake of brevity) for a.e. t ∈ (0,T),x ∈Ω. Further, it follows that \(|w^{j}-w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}=(1-\eta _{j})|w|_{L^{2}(0, T; {H_{0}^{1}}({\Omega }))}\rightarrow 0\) as j →∞, and hence proves this case i = 2 for the a statement and an analogous argument can be used to prove i = 3.
We now consider b. Since \(\tau \in [t_{n-1}^{N}, {t_{n}^{N}})\) is constant, \(\lim _{N\rightarrow \infty }t_{n-1}^{N}=\tau \) and \(\phi ^{N}={\Phi }(t^{N}_{n-1}, u^{N}_{n-1})={\Phi }(t^{N}_{n-1}, \tilde {u}^{N}(t^{N}_{n-1}))\). By (22) we have ϕN → ϕ in L∞(Ω) and in addition ϕN,ϕ ≥ ν > 0 a.e. in Ω. These are the conditions in (68) (with Q exchanged by Ω), and using the same argument we can prove that given a monotonically increasing sequence {ηj} in (0,1) with \(\lim _{j\rightarrow \infty }\eta _{j}= 1\), then wj = ηjw satisfies \(w^{j}\in \mathbf {K}(\phi ^{N(\eta _{j})})\), provided that w ∈K(ϕ), and wj → w in \({H_{0}^{1}}({\Omega })\). This proves the i = 1 case and analogous modifications of the argument in a. proves the cases concerning i = 2 and i = 3. □
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Hintermüller, M., Rautenberg, C.N. & Strogies, N. Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities with Gradient Constraints. Set-Valued Var. Anal 27, 433–468 (2019). https://doi.org/10.1007/s11228-018-0489-0
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DOI: https://doi.org/10.1007/s11228-018-0489-0
Keywords
- Quasi-variational inequality
- Gradient constraint
- Dissipative and non-dissipative processes
- Variable splitting solver