On the convergence of a multigrid method for Moreau-regularized variational inequalities of the second kind

Abstract

We analyze the behavior of a multigrid algorithm for variational inequalities of the second kind with a Moreau-regularized nondifferentiable term. First, we prove a theorem summarizing the properties of the Moreau regularization of a convex, proper, and lower semicontinuous functional that is used in the rest of the paper. We prove that the solution of the regularized problem converges to the solution of the initial problem when the regularization parameter approaches zero. To give a procedure of explicit writing of the Moreau regularization of a convex and lower semicontinuous functional, we have constructed the Moreau regularization for two problems with a scalar unknown taken from the literature and also, for a contact problem with Tresca friction. These functionals are of an integral form and we prove some propositions giving general conditions for which the functionals of this type are lower semicontinuous, proper, and convex. To solve the regularized problem, which is a variational inequality of the first kind, we use a standard multigrid method for two-sided obstacle problems. The numerical experiments have showed a high accuracy and a very good convergence of the method even for values of the regularization parameter close to zero. In view of these results, we think that the proposed method can be an alternative to the existing multigrid methods for the variational inequalities of the second kind.

Appendix. Discrete approach of calculating the Moreau regularization

If the argument of φ is a scalar function, let us consider that

$$ \varphi(v)=\sum\limits_{k=1}^{n_{p}} e_{k}\phi(v(x_{k})) \text{ for any } v=(v(x_{1}),\ldots,v(x_{n_{p}}))\in \mathbf{R}^{n_{p}} $$

(A.1)

where e_k, k = 1,…,n_p, are some nonnegative real constants and $\phi \text { : }\mathbf {R}\to \overline {\mathbf {R}}$. Similarly with Proposition 4.1, we have

Proposition A.1

If$\phi \text { : }\mathbf {R}\to {\overline {\mathbf {R}}}$isa lower semicontinuous, proper and convex function,then$\varphi \text { : }\mathbf {R}^{n_{p}}\to \overline {\mathbf {R}}$definedin (A.1) is a lower semicontinuous, proper and convex functional.

Proof

Evidently, since ϕ is proper and convex, them φ is also a proper and convex functional. Now, let $v_{n}=\{(v^{n}(x_{1}),\ldots ,v^{n}(x_{n_{p}}))\}_{n}\subset \mathbf {R}^{n_{p}}$, $v=(v(x_{1}),\ldots ,v(x_{n_{p}}))\in \mathbf {R}^{n_{p}}$ and ψ ∈R such that v_n → v in $\mathbf {R}^{n_{p}}$, as n →∞, and φ(v_n) ≤ ψ. Then, vⁿ(x_k) → v(x_k) as n →∞, for all k = 1,…,n_p, and, since e_k ≥ 0, k = 1,…,n_p and ϕ is lower semicontinuous, we have

$$ \begin{array}{@{}rcl@{}} &&\varphi(v)= {\sum}_{k=1}^{n_{p}} e_{k}\phi(v(x_{k}))\le {\sum}_{k=1}^{n_{p}} e_{k}\liminf_{n\to\infty}\phi(v^{n}(x_{k})) \\ &&\le \liminf_{n\to\infty}{\sum}_{k=1}^{n_{p}} e_{k}\phi(v^{n}(x_{k}))=\liminf_{n\to\infty} \varphi(v^{n})\le \psi \end{array} $$

i.e., φ is lower semicontinuous. □

Also, with a reasoning similar with that in the case of L²(Ω) we get

$$ \begin{array}{@{}rcl@{}} &&\partial\varphi(u(x_{1}),\ldots,u(x_{n_{p}})) =\left\{(e_{1}\phi^{\prime}_{u(x_{1})},\ldots,e_{n_{p}}\phi^{\prime}_{u(x_{n_{p}})})\in \mathbf{R}^{n_{p}} \text{ : } \phi^{\prime}_{u(x_{k})}\in\mathbf{R},\right. \\ &&\left. \phi^{\prime}_{u(x_{k})}(z_{k}-u(x_{k}))\le \phi(z_{k})-\phi(u(x_{k})) \text{ for any }z_{k}\in \mathbf{R}, k=1,\ldots, n_{p}\right\} \end{array} $$

and, evidently, the value of a subgradient $\partial \varphi _{(u(x_{1}),\ldots ,u(x_{n_{p}}))}\in \partial \varphi (u(x_{1}),\ldots ,u(x_{n_{p}}))$ at a point $(z_{1},\ldots ,z_{n_{p}})\in \mathbf {R}^{n_{p}}$ is written as

$$ \partial\varphi_{(u(x_{1}),\ldots,u(x_{n_{p}}))}(z_{1},\ldots,z_{n_{p}})=\sum\limits_{k=1}^{n_{p}} e_{k} \phi^{\prime}_{u(x_{k})} z_{k} $$

Similarly, if the argument of φ is a vectorial function, we consider $\varphi \text { : }({\mathbf {R}^{\mathbf {d}}})^{n_{p}}\to \overline {\mathbf {R}}$ and

$$ \begin{array}{@{}rcl@{}} &&\displaystyle\varphi(\boldsymbol{v}(x_{1}),\ldots,\boldsymbol{v}(x_{n_{p}}))=\sum\limits_{k=1}^{n_{p}} e_{k}\phi(\boldsymbol{v}(x_{k}))\\ &&\text{for any } \boldsymbol{v}=(\boldsymbol{v}(x_{1}),\ldots,\boldsymbol{v}(x_{n_{p}}))\in ({\mathbf{R}^{\mathbf{d}}})^{n_{p}} \end{array} $$

(A.2)

Remark A.1

Similarly with the above scalar case, we can prove that if $\phi \text { : }{\mathbf {R}^{\mathbf {d}}}\to \overline {\mathbf {R}}$ is a proper, lower semicontinuous, and convex functional which is continuous on its effective domain D(ϕ), then φ defined in (A.2) is proper, lower semicontinuous, and convex.

Also, we have

$$ \begin{array}{@{}rcl@{}} &&\partial\varphi(\boldsymbol{u}(x_{1}),\ldots,\boldsymbol{u}(x_{n_{p}})) =\left\{(e_{1}\boldsymbol{\phi}^{\prime}_{\boldsymbol{u}(x_{1})},\ldots,e_{n_{p}}\boldsymbol{\phi}^{\prime}_{\boldsymbol{u}(x_{n_{p}})})\in ({\mathbf{R}^{\mathbf{d}}})^{n_{p}} \text{ : }\right. \\ &&\boldsymbol{\phi}^{\prime}_{\boldsymbol{u}(x_{k})}\in\mathbf{R}^{d}, \boldsymbol{\phi}^{\prime}_{\boldsymbol{u}(x_{k})}\cdot(\boldsymbol{z}_{k}-\boldsymbol{u}(x_{k}))\le \phi(\boldsymbol{z}_{k})-\phi(\boldsymbol{u}(x_{k})) \\ &&\left. \text{for any }\boldsymbol{z}_{k}\in \mathbf{R}^{d}, k=1,\ldots, n_{p}\right\} \end{array} $$

and the value of a subgradient $\partial \varphi _{(\boldsymbol {u}(x_{1}),\ldots ,\boldsymbol {u}(x_{n_{p}}))}\in \partial \varphi (\boldsymbol {u}(x_{1}),\ldots ,\boldsymbol {u}(x_{n_{p}}))$ at a point $(\boldsymbol {z}_{1},\ldots ,\boldsymbol {z}_{n_{p}})\in (\mathbf {R}^{d})^{n_{p}}$ is written as

$$ \partial\varphi_{(\boldsymbol{u}(x_{1}),\ldots,\boldsymbol{u}(x_{n_{p}}))}(\boldsymbol{z}_{1},\ldots,\boldsymbol{z}_{n_{p}})=\sum\limits_{k=1}^{n_{p}} e_{k} (\boldsymbol{\phi}^{\prime}_{\boldsymbol{u}(x_{k})}\cdot \boldsymbol{z}_{k}) $$

In the following, for completeness, we write the functionals φ_λ and $\varphi ^{\prime }_{\lambda }$ in the three examples in Section 4 when the functional φ is written as in (A.1) or (A.2).

If φ in Example 1 is written as in (A.1), then

$$ \varphi^{\prime}_{\lambda}(u)=(e_{1}\phi^{\prime}_{\lambda}(u(x_{1})),\ldots,e_{n_{p}}\phi^{\prime}_{\lambda }(u(x_{n_{p}}))) $$

where

$$ \begin{array}{@{}rcl@{}} \phi^{\prime}_{\lambda }(u(x_{k}))= \left\{ \begin{array}{l} \frac{1}{1+\lambda a_{1}}[a_{1}(u(x_{k})-\theta_{0})-s_{1}]\text{ if }u(x_{k})-\theta_{0}<-\lambda s_{1} \\ \frac{1}{1+\lambda a_{2}}[a_{2}(u(x_{k})-\theta_{0})+s_{2}]\text{ if }u(x_{k})-\theta_{0}>\lambda s_{2} \\ \frac{u(x_{k})-\theta_{0}}{\lambda}\text{ if }-\lambda s_{1}\le u(x_{k})-\theta_{0}\le\lambda s_{2} \end{array} \right. \end{array} $$

for k = 1,…,n_p. Also,

$$ \begin{array}{@{}rcl@{}} \varphi_{\lambda}(u)&=&{\sum}_{u(x_{k})-\theta_{0}<-\lambda s_{1}} e_{k}\left[\frac{\lambda}{2}\left( \frac{a_{1}(u(x_{k})-\theta_{0})-s_{1}}{1+\lambda a_{1}}\right)^{2} \right. \\ &&\left. + \frac{a_{1}}{2}\left( \frac{u(x_{k})-\theta_{0}+\lambda s_{1}}{1+\lambda a_{1}}\right)^{2} -s_{1}\frac{u(x_{k})-\theta_{0}+\lambda s_{1}}{1+\lambda a_{1}}\right] \\ &&+{\sum}_{u(x_{k})-\theta_{0}>\lambda s_{2}} e_{k}\left[\frac{\lambda}{2}\left( \frac{a_{2}(u(x_{k})-\theta_{0})+s_{2}}{1+\lambda a_{2}}\right)^{2} \right. \\ &&\left. +\frac{a_{2}}{2}\left( \frac{u(x_{k})-\theta_{0}-\lambda s_{2}}{1+\lambda a_{2}}\right)^{2} +s_{2}\frac{u(x_{k})-\theta_{0}-\lambda s_{2}}{1+\lambda a_{2}}\right] \\ &&+ \frac{1}{2\lambda}{\sum}_{-\lambda s_{1}\le u(x_{k})-\theta_{0}\le\lambda s_{2}}e_{k}(u(x_{k})-\theta_{0})^{2} \end{array} $$

(A.3)

In the case of Example 2, $\varphi ^{\prime }_{\lambda }(u)$ has the form in the previous example with

$$ \phi^{\prime}_{\lambda }(u(x_{k}))=\left\{ \begin{array}{ll} 2&\text{ if }u(x_{k})> 1+2\lambda \\ \frac{u(x_{k})-1}{\lambda}&\text{ if }1+\lambda\le u(x_{k})\le 1+2\lambda \\ \frac{1}{2}\left( \sqrt{\lambda^{2}+4u(x_{k})}-\lambda\right)&\text{ if }0\le u(x_{k})<1+\lambda \\ \frac{u(x_{k})}{\lambda}&\text{ if }u(x_{k})< 0 \end{array} \right. $$

for k = 1,…,n_p, and

$$ \begin{array}{@{}rcl@{}} \varphi_{\lambda}(u)&=&{\sum}_{u(x_{k})<0}e_{k}\frac{u(x_{k})^{2}}{2\lambda}+\\ &&{\sum}_{0\le u(x_{k})<1+\lambda}e_{k}\left( \sqrt{\lambda^{2}+4u(x_{k})}-\lambda\right)^{2}\cdot \\ &&\left[\frac{\lambda}{8}+ \frac{1}{12}(\sqrt{\lambda^{2}+4u(x_{k})}-\lambda)\right]\\ &&+ {\sum}_{1+\lambda\le u(x_{k})\le 1+2\lambda}e_{k}\left[\frac{(u(x_{k})-1)^{2}}{2\lambda}+\frac{2}{3}\right] \\ &&+{\sum}_{u(x_{k})> 1+2\lambda}e_{k}\left[2u(x_{k})-2\lambda-\frac{4}{3}\right] \end{array} $$

(A.4)

Finally, if φ in Example 3 is written as in (A.1), i.e.,

$$ \varphi(\boldsymbol{u}(x_{1}),\ldots,\boldsymbol{u}(x_{n_{p}}))=\sum\limits_{k=1}^{n_{p}} e_{k} f(x_{k})\phi(\boldsymbol{u}(x_{k})) $$

then

$$ \varphi^{\prime}_{\lambda}(\boldsymbol{u})=(\boldsymbol{\phi}^{\prime}_{\lambda }(\boldsymbol{u}(x_{1})),\ldots,\phi^{\prime}_{\lambda }(\boldsymbol{u}(x_{n_{p}}))) $$

where

$$ \boldsymbol{\phi}^{\prime}_{\lambda }(\boldsymbol{u}(x_{k}))= \frac{1}{\lambda}\left\{ \begin{array}{ll} \frac{\lambda e_{k}f(x_{k})}{|\boldsymbol{u}_{t}(x_{k})|}\boldsymbol{u}_{t}(x_{k})&\text{ if }|\boldsymbol{u}_{t}(x_{k})|>\lambda e_{k}f(x_{k}) \\ \boldsymbol{u}_{t}(x_{k})&\text{ if }|\boldsymbol{u}_{t}(x_{k})|\le\lambda e_{k} f(x_{k}) \end{array} \right. $$

for k = 1,…,n_p, and

$$ \begin{array}{@{}rcl@{}} \varphi_{\lambda}(\boldsymbol{u})&=&{\sum}_{|\boldsymbol{u}_{t}(x_{k})|>\lambda e_{k}f(x_{k})}e_{k}f(x_{k})\left( |\boldsymbol{u}_{t}(x_{k})|-\frac{\lambda e_{k}f(x_{k})}{2}\right) \\ && +{\sum}_{|\boldsymbol{u}_{t}(x_{k})|\le\lambda e_{k}f(x_{k})}\frac{|\boldsymbol{u}_{t}(x_{k})|^{2}}{2\lambda} \end{array} $$

(A.5)