8,065 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
Geostrophic balance preserving interpolation in mesh adaptive shallow-water ocean modelling
The accurate representation of geostrophic balance is an essential
requirement for numerical modelling of geophysical flows. Significant effort is
often put into the selection of accurate or optimal balance representation by
the discretisation of the fundamental equations. The issue of accurate balance
representation is particularly challenging when applying dynamic mesh
adaptivity, where there is potential for additional imbalance injection when
interpolating to new, optimised meshes.
In the context of shallow-water modelling, we present a new method for
preservation of geostrophic balance when applying dynamic mesh adaptivity. This
approach is based upon interpolation of the Helmholtz decomposition of the
Coriolis acceleration. We apply this in combination with a discretisation for
which states in geostrophic balance are exactly steady solutions of the
linearised equations on an f-plane; this method guarantees that a balanced and
steady flow on a donor mesh remains balanced and steady after interpolation
onto an arbitrary target mesh, to within machine precision. We further
demonstrate the utility of this interpolant for states close to geostrophic
balance, and show that it prevents pollution of the resulting solutions by
imbalanced perturbations introduced by the interpolation
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
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