173 research outputs found
A lowest-order composite finite element exact sequence on pyramids
Composite basis functions for pyramidal elements on the spaces ,
, and are
presented. In particular, we construct the lowest-order composite pyramidal
elements and show that they respect the de Rham diagram, i.e. we have an exact
sequence and satisfy the commuting property. Moreover, the finite elements are
fully compatible with the standard finite elements for the lowest-order
Raviart-Thomas-N\'ed\'elec sequence on tetrahedral and hexahedral elements.
That is to say, the new elements have the same degrees of freedom on the shared
interface with the neighbouring hexahedral or tetrahedra elements, and the
basis functions are conforming in the sense that they maintain the required
level of continuity (full, tangential component, normal component, ...) across
the interface. Furthermore, we study the approximation properties of the spaces
as an initial partition consisting of tetrahedra, hexahedra and pyramid
elements is successively subdivided and show that the spaces result in the same
(optimal) order of approximation in terms of the mesh size as one would
obtain using purely hexahedral or purely tetrahedral partitions.Comment: 21 page
Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by -decompositions
We propose a new tool, which we call -decompositions, for devising
superconvergent hybridizable discontinuous Galerkin (HDG) methods and
hybridized-mixed methods for linear elasticity with strongly symmetric
approximate stresses on unstructured polygonal/polyhedral meshes.
We show that for an HDG method, when its local approximation space admits an
-decomposition, optimal convergence of the approximate stress and
superconvergence of an element-by-element postprocessing of the displacement
field are obtained. The resulting methods are locking-free.
Moreover, we explicitly construct approximation spaces that admit
-decompositions on general polygonal elements. We display numerical results
on triangular meshes validating our theoretical findings.Comment: 45 pages, 2 figure
A systematic construction of finite element commuting exact sequences
We present a systematic construction of finite element exact sequences with a
commuting diagram for the de Rham complex in one-, two- and three-space
dimensions. We apply the construction in two-space dimensions to rediscover two
families of exact sequences for triangles and three for squares, and to uncover
one new family of exact sequence for squares and two new families of exact
sequences for general polygonal elements. We apply the construction in
three-space dimensions to rediscover two families of exact sequences for
tetrahedra, three for cubes, and one for prisms; and to uncover four new
families of exact sequences for pyramids, three for prisms, and one for cubes.Comment: 37page
Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems
We propose energy-conserving discontinuous Galerkin (DG) methods for
symmetric linear hyperbolic systems on general unstructured meshes. Optimal a
priori error estimates of order are obtained for the semi-discrete scheme
in one dimension, and in multi-dimensions on Cartesian meshes when
tensor-product polynomials of degree are used. A high-order
energy-conserving Lax-Wendroff time discretization is also presented.
Extensive numerical results in one dimension, and two dimensions on both
rectangular and triangular meshes are presented to support the theoretical
findings and to assess the new methods. One particular method (with the
doubling of unknowns) is found to be optimally convergent on triangular meshes
for all the examples considered in this paper. The method is also compared with
the classical (dissipative) upwinding DG method and (conservative) DG method
with a central flux. It is numerically observed for the new method to have a
superior performance for long-time simulations.Comment: 48 page
Parameter-free superconvergent -conforming HDG methods for the Brinkman equations
In this paper, we present new parameter-free superconvergent
H(div)-conforming HDG methods for the Brinkman equations on both simplicial and
rectangular meshes. The methods are based on a velocity
gradient-velocity-pressure formulation, which can be considered as a natural
extension of the H(div)-conforming HDG method (defined on simplicial meshes)
for the Stokes flow [Math. Comp. 83(2014), pp. 1571-1598].
We obtain optimal error estimates in -norms for all the variables in
both the Stokes-dominated regime (high viscosity/permeability ratio) and
Darcy-dominated regime (low viscosity/permeability ratio). We also obtain
superconvergent L^2-estimate of one order higher for a suitable projection of
the velocity error, which is typical for (hybrid) mixed methods for elliptic
problems. Moreover, thanks to H(div)-conformity of the velocity, our velocity
error estimates are independent of the pressure regularity.
Preliminary numerical results on both triangular and rectangular meshes in
two-space dimensions confirm our theoretical predictions.Comment: 20 pages, 0 figure
Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for incompressible flow with moving boundaries and interfaces
We present a class of Arbitrary Lagrangian-Eulerian hybridizable
discontinuous Galerkin methods for the incompressible flow with moving
boundaries and interfaces including two-phase flow with surface tension
Uniform auxiliary space preconditioning for HDG methods for elliptic operators with a parameter dependent low order term
The auxiliary space preconditioning (ASP) technique is applied to the HDG
schemes for three different elliptic problems with a parameter dependent low
order term, namely, a symmetric interior penalty HDG scheme for the scalar
reaction-diffusion equation, a divergence-conforming HDG scheme for a vectorial
reaction-diffusion equation, and a C 0 -continuous interior penalty HDG scheme
for the generalized biharmonic equation with a low order term. Uniform
preconditioners are obtained for each case and the general ASP theory by J. Xu
[21] is used to prove the optimality with respect to the mesh size and
uniformity with respect to the low order parameter.Comment: 22 page
A divergence-free HDG scheme for the Cahn-Hilliard phase-field model for two-phase incompressible flow
We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes
phase field model. The scheme is robust in the convection-dominated regime,
produce a globally divergence-free velocity approximation, and can be
efficiently implemented via static condensation. Two numerical benchmark
problems, namely the rising bubble problem, and the Rayleigh-Taylor instability
problem are used to show the good performance of the proposed scheme.Comment: 15 page
An explicit divergence-free DG method for incompressible flow
We present an explicit divergence-free DG method for incompressible flow
based on velocity formulation only. A globally divergence-free finite element
space is used for the velocity field, and the pressure field is eliminated from
the equations by design. The resulting ODE system can be discretized using any
explicit time stepping methods. We use the third order strong-stability
preserving Runge-Kutta method in our numerical experiments. Our spatial
discretization produces the {\it identical} velocity field as the
divergence-conforming DG method of [Cockburn et al., JSC 2007(31), pp.61-73]
based on a velocity-pressure formulation, when the same DG operators are used
for the convective and viscous parts.
Due to the global nature of the divergence-free constraint, there exist no
local bases for our finite element space. We present a key result on the
efficient implementation of the scheme by identifying the equivalence of the
(dense) mass matrix inversion of the globally divergence-free finite element
space to a standard (hybrid-)mixed Poisson solver. Hence, in each time step, a
(hybrid-)mixed Poisson solver is used, which reflects the global nature of the
incompressibility condition. Since we treat viscosity explicitly for the
Navier-Stokes equation, our method shall be best suited for unsteady
high-Reynolds number flows so that the CFL constraint is not too restrictive.Comment: accepted in CMAM
A high-order HDG method for the Biot's consolidation model
We propose a novel high-order HDG method for the Biot's consolidation model
in poroelasticity. We present optimal error analysis for both the semi-discrete
and full-discrete (combined with temporal backward differentiation formula)
schemes. Numerical tests are provided to demonstrate the performance of the
method.Comment: 18 page
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