A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

Abstract

We investigated an hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work [SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J. Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results. Recently, researchers have investigated discontinuous Galerkin (DG) methods for Dirichlet boundary control problems. We used a hybridizable discontinuous Galerkin (HDG) method for the Poisson equation in [25], and obtained a superlinear convergence rate for the control without using a special mesh or a higher order element. More recently, convection diffusion Dirichlet boundary control problems have gained more and more attention. Benner et al. in [3] used a local discontinuous Galerkin (LDG) method to obtain a sublinear convergence rate for the control. We considered an HDG method and proved optimal superlinear convergence rate for the control in [24] if the regularity of the solution is high, i.e., y ∈ H 1+s (Ω) with s ≥ 1/2. To overcome the difficulty for the low regularity case (0 ≤ s < 1/2), we utilized a special projection operator to get an optimal superlinear convergence rate in [18]; the numerical analysis was more complicated than in [24]. Furthermore, in contrast to [25], we obtained an optimal superliner convergence rate for the control by using a discontinuous higher order (quadratic) element.Although the degrees of freedom of the HDG method are significantly reduced compared to standard mixed methods, DG methods and LDG methods, they are still larger than the degrees of freedom of the CG method. In this work, we use embedded DG (EDG) and interior EDG (IEDG) methods to approximate the solution of the Dirichlet boundary control problem. The EDG and IEDG methods are obtained from the HDG methods, and the global systems both use the same continuous elements; this reduces the number of degrees of freedom considerably. To approximate the control, we use continuous element in the EDG method, and discontinuous elements in the IEDG method. Although the degrees of freedom of IEDG is slightly larger than the EDG method, the IEDG method provides greater flexibility for boundary control problems: we can use different finite element spaces for the control and the state. One possible benefit of the greater flexibility of the IEDG method is that discontinuous elements ...

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

“…For the convection diffusion equation, we used a special interpolation operator to deal with this difficulty in [21,37]. Later, in [19], we used a special trace inequality in the numerical analysis of related embedded DG methods; we also use an improved trace inequality in this work, but the analysis is different since the spaces are not the same as in [19].…”

Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems

“…The main difference between EDG and HDG methods is that the approximation spaces for Lagrange multipliers are continuous or not. Some developments of EDG methods could been found in [11,22,35,39,48]. On the other hand, we mention [10,43] for an embeddedhybridized discontinuous Galerkin method for Stokes and Stokes-Darcy problems.…”

Homogeneous multigrid for embedded discontinuous Galerkin methods

“…EDG schemes and their variants have gained some popularity over the last decade. They have, for example, been successfully applied to advection-diffusion [FS17], Stokes [RW20], Euler and Navier-Stokes equations [PNC11,NPC15], distributed optimal control for elliptic problems [ZZS18], Dirichlet boundary control for advection-diffusion [CFSZ19], and compared to stabilized, residual-based finite elements [Kam16]. However, to the best of our knowledge, no multigrid method is available for EDG schemes.…”