A variety of physical and engineering problems develop
dynamically singular or nearly singular solutions in fairly
localized regions. In these problems, we are only interested in
high resolutions in fairly small solution domain. With uniform
meshes, the amount of computational time is too large to enable
us to obtain useful numerical approximations, particular in
multi-dimensions. Therefore, developing effective and robust
moving mesh methods for these problems becomes necessary.
Successful implementation of the adaptive strategy can increase
the accuracy of the numerical approximations and also decrease
the computational cost. In this thesis, we will investigate a
class of moving mesh method which is based on harmonic mapping.
A logical (or computational) domain is used as a reference and
the mesh moving is implemented according to an appropriate
domain transformation. With some traditional moving mesh
methods, the meshes in physical domain may be tangled. To avoid
this, the transformations are constructed based on
harmonic mapping. A good feature of the adaptive methods based
on harmonic mapping is that existence, uniqueness and
non-singularity for the continuous map can be guaranteed from
the theory of harmonic maps. Such theoretical guarantees are
rare in the field of adaptive mesh generation.
In practice, there are three types of adaptive methods using
finite element approach, namely the h-method,
p-method, and r-method (i.e. moving mesh method).
In the h-method, the overall method contains two parts,
i.e. a solution algorithm and a mesh selection algorithm. These
two parts are independent in the sense that the change of the
underlying partial differential equations (PDEs) will affect
the first part only. However, in some of the existing
r-method, these two parts are strongly associated with
each other and as a result any change of the PDEs will result
in the rewriting of the whole code. In this work, we will
propose a moving mesh method which also contains two
parts, a solution algorithm and a mesh-redistribution
algorithm. Our efforts are to keep the advantages for the
r-method (e.g., keep the number of nodes unchanged) and
for the h-method(e.g., the two parts in the code are
independent). The mesh-moving is a procedure of
iteration to construct the harmonic map between the
physical mesh and the logical mesh. Each iteration step is to
move the mesh "closer" to the harmonic map. A new scheme
to interpolate the approximate solution from the old mesh into
the new mesh is designed.
The numerical schemes are applied to a number of test problems
in two- and three-dimensions. It is observed that the
mesh-redistribution strategy
based on the harmonic maps adapts the mesh extremely well to
the solution without producing skew elements for the
multi-dimension computations.
There have been very few moving mesh results for
three-dimensional problems. One of the main difficulties in
dealing with the 3D problems is the treatment of the boundary
grid re-distribution. In 2D, this difficulty can be handled by
solving 1-D moving mesh equations on boundaries. However, the
extension of this boundary grid redistribution technique to 3D
is very difficult. In order to handle this problem, we will
solve a constrained optimization problem that links the
interior and boundary points as whole. It turns out that under
certain conditions this scheme can be implemented in three
dimensions. The mesh generated by the new scheme has higher
quality than that generated by moving mesh methods with fixed
boundaries.
We also applied the moving mesh method to some variational
inequality and elliptical control problems. The key idea is to
construct the monitor function by using appropriate a
posterior error estimates. At the heart of any adaptive
finite element refinement schemes are some appropriate a
posteriori error estimators. The decision of whether further
refinement of meshes is necessary is based on the estimate of
the discretization error. If further refinement is to be
performed then the a posteriori error estimators are used as a
guide as to show the refinement might be accomplished most
efficiently. Now the natural question is can we use them as
monitor functions in the moving mesh methods? One of the
objectives of this thesis is to address the above question. It
is shown that the moving mesh methods with appropriate monitor
functions can effectively solve elliptic obstacle problems and
elliptic control problems.
Our recent works on moving mesh method are its application to
imcompressilbe fluid and DG method.
Some numerical results can be found at the top of this page.