2,263 research outputs found
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
Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes
In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments
Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems
This paper analyzes the local properties of the symmetric interior penalty upwind
discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed
by linear reaction-advection-diffusion equations with distributed controls. The theoretical and
numerical results presented in this paper show that for advection-dominated problems the convergence
properties of the SIPG discretization can be superior to the convergence properties of stabilized
finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For
example, we show that for a small diffusion parameter the SIPG method is optimal in the interior
of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the
existence of boundary layers can pollute the numerical solution of optimal control problems everywhere
even into domains where the solution is smooth and, as a consequence, in general reduces
the convergence rates to only first order. In order to prove the nice convergence properties of the
SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG
discretization for single advection-dominated equations by showing that the size of the numerical
boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter.
As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into
domains of smoothness in optimal control problems. This favorable property of the SIPG method is
due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods,
while for SUPG methods strong imposition of boundary conditions is more conventional. The
importance of the weak treatment of boundary conditions for the solution of advection dominated
optimal control problems with distributed controls is also supported by our numerical results
Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes
A discontinuous Galerkin (dG) method for the numerical solution of
initial/boundary value multi-compartment partial differential equation (PDE)
models, interconnected with interface conditions, is presented and analysed.
The study of interface problems is motivated by models of mass transfer of
solutes through semi-permeable membranes. More specifically, a model problem
consisting of a system of semilinear parabolic advection-diffusion-reaction
partial differential equations in each compartment, equipped with respective
initial and boundary conditions, is considered. Nonlinear interface conditions
modelling selective permeability, congestion and partial reflection are applied
to the compartment interfaces. An interior penalty dG method is presented for
this problem and it is analysed in the space-discrete setting. The a priori
analysis shows that the method yields optimal a priori bounds, provided the
exact solution is sufficiently smooth. Numerical experiments indicate agreement
with the theoretical bounds and highlight the stability of the numerical method
in the advection-dominated regime
An advection-robust Hybrid High-Order method for the Oseen problem
In this work, we study advection-robust Hybrid High-Order discretizations of
the Oseen equations. For a given integer , the discrete velocity
unknowns are vector-valued polynomials of total degree on mesh elements
and faces, while the pressure unknowns are discontinuous polynomials of total
degree on the mesh. From the discrete unknowns, three relevant
quantities are reconstructed inside each element: a velocity of total degree
, a discrete advective derivative, and a discrete divergence. These
reconstructions are used to formulate the discretizations of the viscous,
advective, and velocity-pressure coupling terms, respectively. Well-posedness
is ensured through appropriate high-order stabilization terms. We prove energy
error estimates that are advection-robust for the velocity, and show that each
mesh element of diameter contributes to the discretization error with
an -term in the diffusion-dominated regime, an
-term in the advection-dominated regime, and
scales with intermediate powers of in between. Numerical results complete
the exposition
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