The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data. Furthermore, the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces H and V , with V ⊂ H, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches

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In the first part of this work, we analyzed an unconstrained Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D convex polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory.

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.

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L 2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.AMS Subject Classification. 49J20 and 65N30.The dates will be set by the publisher.