Q. Du
Qiang's Research Gallery 8                
    Adaptive computation


Adaptive algorthms are making many large scale computaional problems affortable with the present day computing technology.
Recently, we have been designing adaptive algorithms for the solution of a variety of applications problems in materials sciences and biological sciences, for example, those associated with an evolving interface.
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).
    map between physical and computational domains
    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.
    mesh motion for 2d microstructure   mesh motion for 2d microstructure

List of collaborators: M.Gunzburger, L. Ju, D. Wang, Z. Huang, J.Zhang, W. Feng, L.Chen, P.Yu, S. Hu

Some references:

Contact  Qiang Du  2006-09-08