10,057 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
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
We derive a posteriori error estimates for the hybridizable discontinuous
Galerkin (HDG) methods, including both the primal and mixed formulations, for
the approximation of a linear second-order elliptic problem on conforming
simplicial meshes in two and three dimensions.
We obtain fully computable, constant free, a posteriori error bounds on the
broken energy seminorm and the HDG energy (semi)norm of the error. The
estimators are also shown to provide local lower bounds for the HDG energy
(semi)norm of the error up to a constant and a higher-order data oscillation
term. For the primal HDG methods and mixed HDG methods with an appropriate
choice of stabilization parameter, the estimators are also shown to provide a
lower bound for the broken energy seminorm of the error up to a constant and a
higher-order data oscillation term. Numerical examples are given illustrating
the theoretical results
An optimal adaptive Fictitious Domain Method
We consider a Fictitious Domain formulation of an elliptic partial
differential equation and approximate the resulting saddle-point system using
an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the
approximation of an elliptic problems performed using adaptive finite element
methods. We prove that the overall method converges with the best possible rate
and illustrate numerically our theoretical findings
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