3,532 research outputs found
Mixed Mimetic Spectral Element Method for Stokes Flow: A Pointwise Divergence-Free Solution
In this paper we apply the recently developed mimetic discretization method
to the mixed formulation of the Stokes problem in terms of vorticity, velocity
and pressure. The mimetic discretization presented in this paper and in [50] is
a higher-order method for curvilinear quadrilaterals and hexahedrals.
Fundamental is the underlying structure of oriented geometric objects, the
relation between these objects through the boundary operator and how this
defines the exterior derivative, representing the grad, curl and div, through
the generalized Stokes theorem. The mimetic method presented here uses the
language of differential -forms with -cochains as their discrete
counterpart, and the relations between them in terms of the mimetic operators:
reduction, reconstruction and projection. The reconstruction consists of the
recently developed mimetic spectral interpolation functions. The most important
result of the mimetic framework is the commutation between differentiation at
the continuous level with that on the finite dimensional and discrete level. As
a result operators like gradient, curl and divergence are discretized exactly.
For Stokes flow, this implies a pointwise divergence-free solution. This is
confirmed using a set of test cases on both Cartesian and curvilinear meshes.
It will be shown that the method converges optimally for all admissible
boundary conditions
The choice of spectral element basis functions in domains with an axis of symmetry
AbstractNew spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared
Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
We formulate hydrodynamic equations and spectrally accurate numerical methods
for investigating the role of geometry in flows within two-dimensional fluid
interfaces. To achieve numerical approximations having high precision and level
of symmetry for radial manifold shapes, we develop spectral Galerkin methods
based on hyperinterpolation with Lebedev quadratures for -projection to
spherical harmonics. We demonstrate our methods by investigating hydrodynamic
responses as the surface geometry is varied. Relative to the case of a sphere,
we find significant changes can occur in the observed hydrodynamic flow
responses as exhibited by quantitative and topological transitions in the
structure of the flow. We present numerical results based on the
Rayleigh-Dissipation principle to gain further insights into these flow
responses. We investigate the roles played by the geometry especially
concerning the positive and negative Gaussian curvature of the interface. We
provide general approaches for taking geometric effects into account for
investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure
Stabilized mixed approximation of axisymmetric Brinkman flows
This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions
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