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

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

Mini-Workshop: Finite Elements and Layer Adapted Meshes

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The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions

In this paper, we consider the symmetric interior penalty Galerkin (SIPG) method which is one of Discontinuous Galerkin Methods for the Dirichlet optimal control problems governed by linear advection-diffusion-reaction equation on a convex polygonal domain and the difficulties which we faced while solving this problem numerically. Since standard Galerkin methods have failed when the boundary layers have occurred and advection diffusion has dominated, these difficulties can occur in the cases of higher order elements and non smooth Dirichlet data in using standard finite elements. We find the most convenient natural setting of Dirichlet boundary control problem for the Laplacian and the advection diffusion reaction equations.After converting the continuous problem to an optimization problem, we solve it by "discretize-then-optimize" approach. In final, we estimate the optimal priori error estimates in suitable norms of the solutions and then support the result and the features of the method with numerical examples on the different kinds of domain

An Optimal EDG Method for Distributed Control of Convection Diffusion PDEs

We propose an embedded discontinuous Galerkin (EDG) method to approximate the solution of a distributed control problem governed by convection diffusion PDEs, and obtain optimal a priori error estimates for the state, dual state, their uxes, and the control. Moreover, we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide. Numerical results confirm our theoretical results

Space-time unfitted finite element methods for time-dependent problems on moving domains

We propose a space-time scheme that combines an unfitted finite element method in space with a discontinuous Galerkin time discretisation for the accurate numerical approximation of parabolic problems with moving domains or interfaces. We make use of an aggregated finite element space to attain robustness with respect to the cut locations. The aggregation is performed slab-wise to have a tensor product structure of the space-time discrete space, which is required in the numerical analysis. We analyse the proposed algorithm, providing stability, condition number bounds and anisotropic \emph{a priori} error estimates. A set of numerical experiments confirm the theoretical results for a parabolic problem on a moving domain. The method is applied for a mass transfer problem with changing topology

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years