[Paper Review] A Superconvergent Hybridizable Discontinuous Galerkin Method for Dirichlet Boundary Control of Elliptic PDEs

Weiwei Hu, Jiguang Shen+3 more|arXiv (Cornell University)|Dec 8, 2017

Advanced Numerical Methods in Computational Mathematics29 references3 citations

TL;DR

This paper proposes a hybridizable discontinuous Galerkin (HDG) method for solving Dirichlet boundary control problems governed by elliptic PDEs. It establishes a superlinear convergence rate of order $ O(h^{3/2- u}) $ for the optimal control in 2D convex polygonal domains under suitable regularity assumptions, validated by 2D and 3D numerical experiments.

ABSTRACT

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

Motivation & Objective

• Address the numerical challenges in solving Dirichlet boundary control problems with low-regularity solutions on polyhedral domains.
• Overcome variational formulation difficulties arising from non-standard boundary conditions in the state equation.
• Develop a high-order, efficient discretization method that reduces degrees of freedom while maintaining accuracy.
• Establish optimal a priori error estimates, particularly for the control variable, under realistic regularity assumptions.
• Demonstrate the method’s robustness and convergence behavior through 2D and 3D numerical experiments.

Proposed method

• Formulate the Dirichlet boundary control problem as a mixed system involving state, adjoint state, and flux variables using a weak formulation.
• Apply a hybridizable discontinuous Galerkin (HDG) method to discretize the state and adjoint equations, enabling local elimination of element-based degrees of freedom.
• Use Raviart-Thomas-type finite element spaces for the fluxes and discontinuous polynomials for the state and adjoint variables.
• Construct a local solver via block matrix factorization, enabling efficient solution of the global system through static condensation.
• Ensure the resulting linear system is symmetric and positive definite, allowing for efficient iterative solvers.
• Leverage the HDG structure to achieve superconvergence in the control variable by exploiting the structure of the numerical fluxes and postprocessing.

Experimental results

Research questions

• RQ1 Can HDG methods achieve optimal convergence rates for the optimal control in Dirichlet boundary control problems with low-regularity solutions?
• RQ2 Does the HDG method maintain high accuracy and efficiency when applied to elliptic PDEs with nonstandard variational formulations?
• RQ3 What is the convergence rate of the control variable in 2D convex polygonal domains under standard regularity assumptions?
• RQ4 How does the HDG method compare to standard mixed finite element methods in terms of degrees of freedom and convergence behavior?
• RQ5 Can the HDG framework be extended to more complex problems involving convection-dominated flows or discontinuities?

Key findings

• For a 2D convex polygonal domain with $ y_d otin H^1(ar{ abla}) $, the optimal control converges at a superlinear rate of $ O(h^{3/2- u}) $ for any $ u > 0 $, under appropriate regularity assumptions.
• The state and adjoint state converge at $ O(h^{3/2- u}) $, while their fluxes converge at $ O(h^{1- u}) $, indicating optimal convergence for the mixed variables.
• In 3D, the numerical experiments confirm convergence rates consistent with the theoretical estimates: $ O(h^{3/2- u}) $ for the state and adjoint state, and $ O(h^{1- u}) $ for the fluxes.
• The control error converges at approximately $ O(h^{1.5}) $ in 2D and $ O(h^{1.3}) $ in 3D, approaching the theoretical superlinear rate.
• The HDG method significantly reduces the number of globally coupled degrees of freedom compared to standard mixed methods, due to static condensation.
• The local solver is computationally efficient due to block-diagonal structure and parallelizable inversion of small blocks, enabling scalable implementation.