51 research outputs found
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to IP methods
In this article, an abstract framework for the error analysis of
discontinuous Galerkin methods for control constrained optimal control problems
is developed. The analysis establishes the best approximation result from a
priori analysis point of view and delivers reliable and efficient a posteriori
error estimators. The results are applicable to a variety of problems just
under the minimal regularity possessed by the well-posed ness of the problem.
Subsequently, applications of interior penalty methods for a boundary
control problem as well as a distributed control problem governed by the
biharmonic equation subject to simply supported boundary conditions are
discussed through the abstract analysis. Numerical experiments illustrate the
theoretical findings. Finally, we also discuss the variational discontinuous
discretization method (without discretizing the control) and its corresponding
error estimates.Comment: 23 pages, 5 figures, 1 tabl
A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems
In this article, we design and analyze a Hybrid High-Order (HHO) finite
element approximation for a class of strongly nonlinear boundary value
problems. We consider an HHO discretization for a suitable linearized problem
and show its well-posedness using the Gardings type inequality. The essential
ingredients for the HHO approximation involve local reconstruction and
high-order stabilization. We establish the existence of a unique solution for
the HHO approximation using the Brouwer fixed point theorem and contraction
principle. We derive an optimal order a priori error estimate in the discrete
energy norm. Numerical experiments are performed to illustrate the convergence
histories.Comment: arXiv admin note: substantial text overlap with arXiv:2110.1557
An hp-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type
In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H1 norm and L2 norm which are optimal in h, suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results
Error analysis of discontinuous Galerkin methods for Stokes problem under minimal regularity
In this article, we analyze several discontinuous Galerkin methods (DG) for the
Stokes problem under the minimal regularity on the solution. We assume that the velocity
u belongs to [H1 0 ()]d and the pressure p 2 L2 0 (). First, we analyze standard DG methods assuming that the right hand side f belongs to [H¡1() \ L1()]d. A DG method that is well de¯ned for f belonging to [H¡1()]d is then investigated. The methods under study
include stabilized DG methods using equal order spaces and inf-sup stable ones where the pressure space is one polynomial degree less than the velocity space.Preprin
Finite element pressure stabilizations for incompressible flow problems
Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis
Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems
An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a GAyenrding type inequality. Optimal order L (2) norm a priori error estimates are derived for an adjoint consistent interior penalty method
Conforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem
In this article, we propose and analyze conforming and discontinuous Galerkin (DG) finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in R-d (d = 2, 3). For fully discrete conforming method, we use globally continuous and piecewise linear finite element space. Whereas for the fully-discrete DG scheme, we employ piecewise linear finite element space for spatial discretization. The time discretization has been done by using the implicit backward Euler method. We present the error analysis for the conforming and the DG fully discrete schemes and derive an error estimate of optimal order (h+ Delta t) in a certain energy norm defined precisely in the article. The analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that u(t) is an element of L-2 (0, T; L-2(Omega)). The obstacle constraints are incorporated at the Lagrange nodes of the triangular mesh and the analysis exploits the Lagrange interpolation. We present some numerical experiment to illustrate the performance of the proposed methods
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