47 research outputs found
A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations
Convection-diffusion equations arise in a variety of applications such as
particle transport, electromagnetics, and magnetohydrodynamics. Simulation of
the convection-dominated regime for these problems, even with high-fidelity
techniques, is particularly challenging due to the presence of sharp boundary
layers and shocks causing jumps and discontinuities in the solution, and
numerical issues such as loss of the maximum principle in the discretization.
These complications cause instabilities, admitting large oscillations in the
numerical solution when using traditional methods. Drawing connections to the
simplex-averaged finite-element method (S. Wu and J. Xu, 2020), this paper
develops a mimetic finite-difference (MFD) discretization using
exponentially-averaged coefficients to overcome instability of the numerical
solution as the diffusion coefficient approaches zero. The finite-element
framework allows for transparent analysis of the MFD, such as proving
well-posedness and deriving error estimates. Numerical tests are presented
confirming the stability of the method and verifying the error estimates
Exponentially-fitted finite elements for and convection-diffusion problems
This paper presents a novel approach to the construction of the lowest order
and exponentially-fitted finite element
spaces on 3D simplicial mesh for
corresponding convection-diffusion problems. It is noteworthy that this method
not only facilitates the construction of the functions themselves but also
provides corresponding discrete fluxes simultaneously. Utilizing this approach,
we successfully establish a discrete convection-diffusion complex and employ a
specialized weighted interpolation to establish a bridge between the continuous
complex and the discrete complex, resulting in a coherent framework.
Furthermore, we demonstrate the commutativity of the framework when the
convection field is locally constant, along with the exactness of the discrete
convection-diffusion complex. Consequently, these types of spaces can be
directly employed to devise the corresponding discrete scheme through a
Petrov-Galerkin method
The auxiliary space preconditioner for the de Rham complex
We generalize the construction and analysis of auxiliary space
preconditioners to the n-dimensional finite element subcomplex of the de Rham
complex. These preconditioners are based on a generalization of a decomposition
of Sobolev space functions into a regular part and a potential. A discrete
version is easily established using the tools of finite element exterior
calculus. We then discuss the four-dimensional de Rham complex in detail. By
identifying forms in four dimensions (4D) with simple proxies, form operations
are written out in terms of familiar algebraic operations on matrices, vectors,
and scalars. This provides the basis for our implementation of the
preconditioners in 4D. Extensive numerical experiments illustrate their
performance, practical scalability, and parameter robustness, all in accordance
with the theory
Discontinuous Galerkin methods for magnetic advection-diffusion problems
We devise and analyze a class of the primal discontinuous Galerkin methods
for the magnetic advection-diffusion problems based on the weighted-residual
approach. In addition to the upwind stabilization, we find a new mechanism
under the vector case that provides more flexibility in constructing the
schemes. For the more general Friedrichs system, we show the stability and
optimal error estimate, which boil down to two core ingredients -- the weight
function and the special projection -- that contain information of advection.
Numerical experiments are provided to verify the theoretical results
Discretisations and Preconditioners for Magnetohydrodynamics Models
The magnetohydrodynamics (MHD) equations are generally known to be difficult
to solve numerically, due to their highly nonlinear structure and the strong
coupling between the electromagnetic and hydrodynamic variables, especially for
high Reynolds and coupling numbers.
In the first part of this work, we present a scalable augmented Lagrangian
preconditioner for a finite element discretisation of the
- formulation of the incompressible viscoresistive MHD
equations. For stationary problems, our solver achieves robust performance with
respect to the Reynolds and coupling numbers in two dimensions and good results
in three dimensions. Our approach relies on specialised parameter-robust
multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme
ensures exactly divergence-free approximations of both the velocity and the
magnetic field up to solver tolerances.
In the second part, we focus on incompressible, resistive Hall MHD models and
derive structure-preserving finite element methods for these equations. We
present a variational formulation of Hall MHD that enforces the magnetic
Gauss's law precisely (up to solver tolerances) and prove the well-posedness of
a Picard linearisation. For the transient problem, we present time
discretisations that preserve the energy and magnetic and hybrid helicity
precisely in the ideal limit for two types of boundary conditions.
In the third part, we investigate anisothermal MHD models. We start by
performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at
a high coupling number by choosing the Rayleigh number in the range
between 0 and as the bifurcation parameter. We study the effect of
the coupling number on the bifurcation diagram and outline how we create
initial guesses to obtain complex solution patterns and disconnected branches
for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174
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Convection in Rotating Spherical Fluid Shell
t has been recently observed that there is a change in the length of day with a period of roughly six years, while a longer period in the change of length of day is attributed to action within Earth’s core, the six year period is theorized to be a product of the Earth’s dynamo generation. It is conjectured that a torsional wave with a six year periodicity, i.e. an oscillation in the azimuthal velocity of the outer core that propagates radially outward could explain the change in the length of the day via conservation of angular momentum. Previous simulations have identified torsional waves in both dynamo simulations and purely hydrodynamical simulations. Here we explore torsional waves as a purely hydrodynamical process, modeling Boussinesq fluids in arotating spherical shell, with no-slip boundary condition. The numerical simulations are doneby solving the Navier-Stokes equation sets using spherical Dedalus, a pseudo-spectral partial differential equation solver. There were no successful identification of torsional waves for the rapidly rotating parameter regime Ek ~10^{-6} and Ra ~ 10^8
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions