2,888 research outputs found
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
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
Compatible finite element methods for numerical weather prediction
This article takes the form of a tutorial on the use of a particular class of
mixed finite element methods, which can be thought of as the finite element
extension of the C-grid staggered finite difference method. The class is often
referred to as compatible finite elements, mimetic finite elements, discrete
differential forms or finite element exterior calculus. We provide an
elementary introduction in the case of the one-dimensional wave equation,
before summarising recent results in applications to the rotating shallow water
equations on the sphere, before taking an outlook towards applications in
three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201
Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
Summation-by-parts (SBP) operators allow us to systematically develop
energy-stable and high-order accurate numerical methods for time-dependent
differential equations. Until recently, the main idea behind existing SBP
operators was that polynomials can accurately approximate the solution, and SBP
operators should thus be exact for them. However, polynomials do not provide
the best approximation for some problems, with other approximation spaces being
more appropriate. We recently addressed this issue and developed a theory for
one-dimensional SBP operators based on general function spaces, coined
function-space SBP (FSBP) operators. In this paper, we extend the theory of
FSBP operators to multiple dimensions. We focus on their existence, connection
to quadratures, construction, and mimetic properties. A more exhaustive
numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their
application will be provided in future works. Similar to the one-dimensional
case, we demonstrate that most of the established results for polynomial-based
multi-dimensional SBP (MSBP) operators carry over to the more general class of
MFSBP operators. Our findings imply that the concept of SBP operators can be
applied to a significantly larger class of methods than is currently done. This
can increase the accuracy of the numerical solutions and/or provide stability
to the methods.Comment: 28 pages, 9 figure
Virtual Element Methods for hyperbolic problems on polygonal meshes
In the present paper we develop the Virtual Element Method for hyperbolic
problems on polygonal meshes, considering the linear wave equations as our
model problem. After presenting the semi-discrete scheme, we derive the
convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a
theoretical analysis on the stability for the fully discrete problem by
comparing the Newmark method and the Bathe method. Finally we show the
practical behaviour of the proposed method through a large array of numerical
tests
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