482 research outputs found
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
Parameter-robust discretization and preconditioning of Biot's consolidation model
Biot's consolidation model in poroelasticity has a number of applications in
science, medicine, and engineering. The model depends on various parameters,
and in practical applications these parameters ranges over several orders of
magnitude. A current challenge is to design discretization techniques and
solution algorithms that are well behaved with respect to these variations. The
purpose of this paper is to study finite element discretizations of this model
and construct block diagonal preconditioners for the discrete Biot systems. The
approach taken here is to consider the stability of the problem in non-standard
or weighted Hilbert spaces and employ the operator preconditioning approach. We
derive preconditioners that are robust with respect to both the variations of
the parameters and the mesh refinement. The parameters of interest are small
time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page
Schnelle Löser für Partielle Differentialgleichungen
This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
We discuss the construction of robust preconditioners for finite element
approximations of Biot's consolidation model in poroelasticity. More precisely,
we study finite element methods based on generalizations of the
Hellinger-Reissner principle of linear elasticity, where the stress tensor is
one of the unknowns. The Biot model has a number of applications in science,
medicine, and engineering. A challenge in many of these applications is that
the model parameters range over several orders of magnitude. Therefore,
discretization procedures which are well behaved with respect to such
variations are needed. The focus of the present paper will be on the
construction of preconditioners, such that the preconditioned discrete systems
are well-conditioned with respect to variations of the model parameters as well
as refinements of the discretization. As a byproduct, we also obtain
preconditioners for linear elasticity that are robust in the incompressible
limit.Comment: 21 page
An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations
We present and analyze a new embedded--hybridized discontinuous Galerkin
finite element method for the Stokes problem. The method has the attractive
properties of full hybridized methods, namely an -conforming
velocity field, pointwise satisfaction of the continuity equation and \emph{a
priori} error estimates for the velocity that are independent of the pressure.
The embedded--hybridized formulation has advantages over a full hybridized
formulation in that it has fewer global degrees-of-freedom for a given mesh and
the algebraic structure of the resulting linear system is better suited to fast
iterative solvers. The analysis results are supported by a range of numerical
examples that demonstrate rates of convergence, and which show computational
efficiency gains over a full hybridized formulation
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