3 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
Mimetic framework on curvilinear quadrilaterals of arbitrary order
In this paper higher order mimetic discretizations are introduced which are
firmly rooted in the geometry in which the variables are defined. The paper
shows how basic constructs in differential geometry have a discrete counterpart
in algebraic topology. Generic maps which switch between the continuous
differential forms and discrete cochains will be discussed and finally a
realization of these ideas in terms of mimetic spectral elements is presented,
based on projections for which operations at the finite dimensional level
commute with operations at the continuous level. The two types of orientation
(inner- and outer-orientation) will be introduced at the continuous level, the
discrete level and the preservation of orientation will be demonstrated for the
new mimetic operators. The one-to-one correspondence between the continuous
formulation and the discrete algebraic topological setting, provides a
characterization of the oriented discrete boundary of the domain. The Hodge
decomposition at the continuous, discrete and finite dimensional level will be
presented. It appears to be a main ingredient of the structure in this
framework.Comment: 69 page