Our efforts include the study of:
- Mesh generation, adaptation, and optimization
Our work in this direction includes CVT based mesh optimization and adaptation,
isotropic and anisotropic mesh generation, boundary recovery.
- Mesh-solver co-adaptation
We advocate the combined effort to adapt mesh and solver jointly so that
the meshes are obtained not only to provide maximum resolution but also
to lead efficient reductions of linear solvers.
- A posteriori error estimation
We carefully analyze and derive various a posteriori error estimators for
a number of nonlinear problems and use them to effectively characterize the
behavior of the numerical solution and to design mesh adaptation procedure
For example, in 3D phase field modeling and simulations of vesicle
membranes, phase field functions are computed in 3d domain mainly to
resolve the 2d vesicle, our computation shows that
the adaptive FEM for vesicle deformation based on residual type
a posteriori error estimates (Du-Zhang 2007) can effectively
reduce the 3D Computation to 2D complexity
- Adaptive spectral methods
We ultize the moving mesh ideas and the high resolutions and
Fast FFT-based implementation of spectral methods to develop
adaptive spectral methods for nonlinear PDEs.
Large gradient of the solution in physical domain
requires high resolution, thus necessitates fine mesh.
For many practical problems in high dimensional space,
it can be prohibitive computationally if the fine mesh is spatially
uniform.
To make grids clustered near physical interface, a map is used between the computational and physical domain so that the grids still remain uniform in the computational domain.
To minimize the overhead, the moving mesh PDEs are solved in a similar manner as the original phase field models via semi-implict integration with time splitting. We name it the Moving Mesh Fourier spectral method (MMFS).
MMFS: mesh evolves with solution to provide maximum resolution
MMFS overcomes the complication of inhomogeneous coefficients due to mesh motion to allow FFT, reduce overall system size and improve accuracy.
In many test runs, we see the performance gain despite of the overhead.
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