90 research outputs found
A superconvergent HDG method for stokes flow with strongly enforced symmetry of the stress tensor
This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types
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
A superconvergent HDG method for stokes flow with strongly enforced symmetry of the stress tensor
The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-018-0855-yThis work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.Peer ReviewedPostprint (author's final draft
Minimal order H(div)-conforming velocity-vorticity approximations for incompressible fluids
We introduce a novel minimal order hybrid Discontinuous Galerkin (HDG) and a
novel mass conserving mixed stress (MCS) method for the approximation of
incompressible flows. For this we employ the -conforming
linear Brezzi-Douglas-Marini space and the lowest order Raviart-Thomas space
for the approximation of the velocity and the vorticity, respectively. Our
methods are based on the physically correct diffusive flux and provide exactly divergence-free discrete velocity
solutions, optimal (pressure robust) error estimates and a minimal number of
coupling degrees of freedom. For the stability analysis we introduce a new
Korn-like inequality for vector-valued element-wise and normal continuous
functions. Numerical examples conclude the work where the theoretical findings
are validated and the novel methods are compared in terms of condition numbers
with respect to discrete stability parameters
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