Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical analysis of a nonsmooth quasilinear elliptic control problem: II. Finite element discretization and error estimates

Christian Clason, Vu Huu Nhu, Arnd Rösch

TL;DR

This work tackles a nonsmooth quasilinear elliptic optimal control problem where the coefficient $a(y)$ is Lipschitz but nondifferentiable at a threshold $\bar t$, leading to a control-to-state map $S$ that is $C^1$ but not $C^2$. The authors develop a rigorous FE discretization framework for the state and adjoint equations, introducing an adjusted linearized problem and a curvature functional decomposition $Q=Q_s+Q_1+Q_2$ to perform error analysis under an explicit second-order sufficient optimality condition. They prove convergence of discrete minimizers to a global continuous minimizer and derive a priori error estimates for variational, piecewise-constant, and continuous piecewise linear discretizations, highlighting rates such as $O(h^{\tfrac{3}{2}-\varepsilon})$ in the general nonsmooth setting and potential superconvergence in smooth cases. A numerical example demonstrates the predicted rates and confirms the theoretical results, while also illustrating the practical impact of the nonsmooth coefficient on convergence. These results provide a principled pathway for reliable numerical optimization in PDEs with nonsmooth nonlinearities and contribute to the understanding of discretization effects in nonsmooth optimal control problems.

Abstract

In this paper, we carry out the numerical analysis of a nonsmooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is not differentiable with respect to the state variable. Despite the lack of differentiability of the nonlinearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class $C^1$ but not of class $C^2$. Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class $C^1$ only. By using an explicit second-order sufficient optimality condition, we prove a priori error estimates for a variational approximation, a piecewise constant approximation, and a continuous piecewise linear approximation of the continuous optimal control problem. The numerical tests confirm these error estimates.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: II. Finite element discretization and error estimates

TL;DR

This work tackles a nonsmooth quasilinear elliptic optimal control problem where the coefficient is Lipschitz but nondifferentiable at a threshold , leading to a control-to-state map that is but not . The authors develop a rigorous FE discretization framework for the state and adjoint equations, introducing an adjusted linearized problem and a curvature functional decomposition to perform error analysis under an explicit second-order sufficient optimality condition. They prove convergence of discrete minimizers to a global continuous minimizer and derive a priori error estimates for variational, piecewise-constant, and continuous piecewise linear discretizations, highlighting rates such as in the general nonsmooth setting and potential superconvergence in smooth cases. A numerical example demonstrates the predicted rates and confirms the theoretical results, while also illustrating the practical impact of the nonsmooth coefficient on convergence. These results provide a principled pathway for reliable numerical optimization in PDEs with nonsmooth nonlinearities and contribute to the understanding of discretization effects in nonsmooth optimal control problems.

Abstract

In this paper, we carry out the numerical analysis of a nonsmooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is not differentiable with respect to the state variable. Despite the lack of differentiability of the nonlinearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class but not of class . Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class only. By using an explicit second-order sufficient optimality condition, we prove a priori error estimates for a variational approximation, a piecewise constant approximation, and a continuous piecewise linear approximation of the continuous optimal control problem. The numerical tests confirm these error estimates.
Paper Structure (11 sections, 33 theorems, 263 equations, 1 table)

This paper contains 11 sections, 33 theorems, 263 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main assumptions and preliminary results
  3. Analysis of the discrete state equation
  4. Numerical analysis of the adjoint state equation
  5. Discretization of the control problem
  6. Convergence of discrete minimizers
  7. Error estimates for discrete minimizers
  8. Numerical example
  9. Conclusions
  10. Verification of a structural assumption and computation of the jump functional
  11. Regularity of solutions to the adjusted linearized state equation

Key Result

Theorem 2.2

Assume that ass:domainass:PC1-func are verified. Then, the control-to-state mapping $S: W^{-1,p}(\Omega) \ni u \mapsto y_u \in W^{1,p}_0(\Omega)$ is of class $C^1$, where $y_u$ is the unique solution to eq:state. Moreover, for any $u,v \in W^{-1,p}(\Omega)$ with $p > 2$ and $y_u:= S(u)$, $z_v:= S'(u Moreover, a number $p_* >2$ exists and satisfies that for any $p \in [2,p_*)$ and for any bounded s

Theorems & Definitions (59)

  • Remark 2.1
  • Theorem 2.2: ClasonNhuRosch_levelset, Thm. 3.2 and cf. ClasonNhuRosch_os2nd, Thms. 3.1 and 3.5
  • Theorem 2.3: ClasonNhuRosch_levelset, Thm. 3.3 and cf. ClasonNhuRosch_os2nd, Lem. 4.1
  • Remark 2.4
  • Theorem 2.5: ClasonNhuRosch_os2nd, Thm. 4.3
  • Remark 2.6
  • Theorem 2.7: explicit second-order sufficient optimality conditions, ClasonNhuRosch_levelset, Thm. 3.23
  • Proposition 2.8: ClasonNhuRosch_levelset, Thm. 3.19
  • Proposition 2.9: ClasonNhuRosch_levelset, Prop. 3.20
  • Theorem 3.1: CasasTroltzsch2011, Thm. 3.1
  • ...and 49 more