TY - THES
AU - Li, Dan
PY - 2010
TI - Numerical solution of the time-harmonic Maxwell equations and incompressible magnetohydrodynamics problems
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - The goal of this thesis is to develop efficient numerical solvers for the time-harmonic
Maxwell equations and for incompressible magnetohydrodynamics problems.
The thesis consists of three components. In the first part, we present a fully
scalable parallel iterative solver for the time-harmonic Maxwell equations in mixed
form with small wave numbers. We use the lowest order Nedelec elements of the
first kind for the approximation of the vector field and standard nodal elements for
the Lagrange multiplier associated with the divergence constraint. The corresponding
linear system has a saddle point form, with inner iterations solved by preconditioned conjugate gradients. We demonstrate the performance of
our parallel solver on problems with constant and variable coefficients with up to
approximately 40 million degrees of freedom. Our numerical results indicate very
good scalability with the mesh size, on uniform, unstructured and locally refined
meshes.
In the second part, we introduce and analyze a mixed finite element method for
the numerical discretization of a stationary incompressible magnetohydrodynamics
problem, in two and three dimensions. The velocity field is discretized using
divergence-conforming Brezzi-Douglas-Marini (BDM) elements and the magnetic
field is approximated by curl-conforming Nedelec elements. Key features of the
method are that it produces exactly divergence-free velocity approximations, and
that it correctly captures the strongest magnetic singularities in non-convex polyhedral
domains. We prove that the energy norm of the error is convergent in the
mesh size in general Lipschitz polyhedra under minimal regularity assumptions,
and derive nearly optimal a-priori error estimates for the two-dimensional case. We
present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional
problems.
Finally, in the third part we investigate preconditioned Krylov iterations for
the discretized stationary incompressible magnetohydrodynamics problems. We
propose a preconditioner based on efficient preconditioners for the Maxwell and
Navier-Stokes sub-systems. We show that many of the eigenvalues of the preconditioned
system are tightly clustered, and hence, rapid convergence is accomplished.
Our numerical results show that this approach performs quite well.
N2 - The goal of this thesis is to develop efficient numerical solvers for the time-harmonic
Maxwell equations and for incompressible magnetohydrodynamics problems.
The thesis consists of three components. In the first part, we present a fully
scalable parallel iterative solver for the time-harmonic Maxwell equations in mixed
form with small wave numbers. We use the lowest order Nedelec elements of the
first kind for the approximation of the vector field and standard nodal elements for
the Lagrange multiplier associated with the divergence constraint. The corresponding
linear system has a saddle point form, with inner iterations solved by preconditioned conjugate gradients. We demonstrate the performance of
our parallel solver on problems with constant and variable coefficients with up to
approximately 40 million degrees of freedom. Our numerical results indicate very
good scalability with the mesh size, on uniform, unstructured and locally refined
meshes.
In the second part, we introduce and analyze a mixed finite element method for
the numerical discretization of a stationary incompressible magnetohydrodynamics
problem, in two and three dimensions. The velocity field is discretized using
divergence-conforming Brezzi-Douglas-Marini (BDM) elements and the magnetic
field is approximated by curl-conforming Nedelec elements. Key features of the
method are that it produces exactly divergence-free velocity approximations, and
that it correctly captures the strongest magnetic singularities in non-convex polyhedral
domains. We prove that the energy norm of the error is convergent in the
mesh size in general Lipschitz polyhedra under minimal regularity assumptions,
and derive nearly optimal a-priori error estimates for the two-dimensional case. We
present a comprehensive set of numerical experiments, which indicate optimal convergence of the proposed method for two-dimensional as well as three-dimensional
problems.
Finally, in the third part we investigate preconditioned Krylov iterations for
the discretized stationary incompressible magnetohydrodynamics problems. We
propose a preconditioner based on efficient preconditioners for the Maxwell and
Navier-Stokes sub-systems. We show that many of the eigenvalues of the preconditioned
system are tightly clustered, and hence, rapid convergence is accomplished.
Our numerical results show that this approach performs quite well.
UR - https://open.library.ubc.ca/collections/24/items/1.0051989
ER - End of Reference