73 research outputs found
HDG methods for Dirichlet boundary control of PDEs
We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for 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. In this thesis, we use an existing HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control in the high regularity case. We also propose a new HDG method to approximate the solution of a Dirichlet boundary control problem governed by a linear elliptic convection diffusion PDE. Although there are many works in the literature on Dirichlet boundary control problems for the Poisson equation, we are not aware of any existing theoretical or numerical analysis works for convection diffusion Dirichlet control problems. We obtainwell-posedness and regularity results for the Dirichlet control problem, and we prove optimal a priori error estimates in 2D for the control in both the high regularity and low regularity cases. As far as the authors are aware, there are no existing comparable results in the literature. Moreover, we present numerical experiments to demonstrate the performance of the HDG methods and illustrate our numerical analysis results --Abstract, page iii
Superconvergent interpolatory HDG methods for reaction diffusion equations I: An HDG method
In our earlier work [8], we approximated solutions of a general class of
scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous
Galerkin (Interpolatory HDG) method. This method reduces the computational cost
compared to standard HDG since the HDG matrices are assembled once before the
time integration. Interpolatory HDG also achieves optimal convergence rates;
however, we did not observe superconvergence after an element-by-element
postprocessing. In this work, we revisit the Interpolatory HDG method for
reaction diffusion problems, and use the postprocessed approximate solution to
evaluate the nonlinear term. We prove this simple change restores the
superconvergence and keeps the computational advantages of the Interpolatory
HDG method. We present numerical results to illustrate the convergence theory
and the performance of the method
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