Description of Current Research

Research Statement

Overview:

My current research works are focused on finding an accurate and efficient solver for the convection-diffusion equations. I decided to work in this area because the convection-diffusion problem is one of the main problems in thermal dynamics and it plays an important role in solving the linearized Navier-Stokes equation. Since many applications in computational fluid dynamics (CFD) simulations require a fast and reliable solver of Navier-Stokes equations, the convection-diffusion equation, as a core component of the linearized Navier-Stokes equation, has to be solved accurately and efficiently. This can become a challenge, especially, when the convection-diffusion problem becomes singularly perturbed at high Reynolds number. In this scenario, the major difficulties in large scale computing for the convection-diffusion problem include:

1. The solutions from the standard Galerkin finite element method (GFEM) suffer from large spurious oscillation when the mesh Peclét number is much greater than one.

2. Due to limited computation resources, the resolution of the numerical solutions on uniform meshes are generally not high enough to provide detailed solution structure in the layer regions, which are more interested in most practical applications.

3. The matrix obtained from FEM discretization of the convection-diffusion equation is highly non-symmetric and nearly positive-indefinite. As a result, stationary iterative solvers, such as Jacobi and Gauss-Seidel, and Krylov subspace methods, including the generalized minimal residual method (GMRES), converge slowly in general.

To overcome these difficulties, one requires 1) a stable discretization scheme such as the streamline diffusion finite element method (SDFEM) to reduce the numerical spurious oscillation, 2) a reliable a posteriori error estimator to estimate the global error and identify regions where the errors are large, 3) a mesh refinement strategy to enhance the mesh resolution locally, and 4) a fast iterative solver on adaptively refined meshes.

Accomplishment:

In my dissertation, the quality of SDFEM solution is reassured. To increase the solution accuracy, a reliable a posteriori error estimator recently proposed by Kay and Silvester is studied and compared with the a posteriori error estimator proposed by Verfurth in 1998. My numerical results indicate that the local lower bounds are sharp for both error estimators and the Verfurth’s error estimator is less reliable than Kay and Silvester’s error estimator is. Therefore, Kay and Silvester’s error estimator is used in the numerical tests of mesh movement and mesh refinement. Although the regular mesh refinement strategy provides a mechanism to increase solution accuracy without largely increasing the computation cost at each refinement step, one may still need too many refinement steps in order to obtain satisfactory resolution in layer regions. A new error-adapted mesh refinement strategy is proposed to relieve this difficulty. By applying the error-adapted mesh refinement strategy to many benchmark problems, the numerical results show great improvement on the accuracy of boundary layers compared to the regular mesh refinement strategy. Furthermore, in some cases, I found that if the boundary layers can be resolved faster, the global accuracy of the approximate solution could be increased significantly. Another benefit from the error-adapted refinement strategy is that the nested mesh structures are automatically maintained and ready for multigrid computation.

For fast convergence of iterative linear solvers, first, some convergence results of both line Gauss-Seidel iteration method and multigrid method are proved for a constant flow benchmark problem on uniform grids, where the mesh Peclét number is greater than 1/h and h is the mesh size. Theoretical analysis in my dissertation not only shows multigrid convergence, but also shows that multigrid converges faster than the line Gauss-Seidel. Numerical results are also consistent with this conclusion. Second, without acquiring uniform meshes and constrains on the mesh Peclét number, the performance of multigrid with standard bilinear interpolation (MG), algebraic multigrid (AMG), and GMRES with MG and AMG preconditioners are evaluated on both uniform meshes and adaptively refined meshes for many benchmark problems including flows with close characteristics. My numerical results show that MG and AMG are good preconditioners of GMRES on both types of meshes. However, if the approximate solutions are solved only on adaptively refined meshes, MG alone can be considered a good linear solver. Finally, new stopping criteria, based on a posteriori estimation, are proposed. The motivation to develop these stopping criteria is based on the following ideas:

1. As long as the number of iterations is large enough to ensure the error between the iterative solution and the exact solution is bounded by the a posteriori error bound on a given mesh, one cannot distinguish the iterative solution and the exact solution from a computable error estimations point of view.

2. If the number of iterations is large enough such that the a posteriori error estimator computed from the iterative solution and the a posteriori error estimator computed from the exact solution are similar, one should also have similar meshes refined from these error estimators for a given element marking strategy.

By using these stopping criteria on the MG solver, the computation time along the adaptive refinement process is almost half of the computation time required when the heuristic stopping criterion, i.e. the residual reduction rate less than 10^-6, is applied on the MG solver. Moreover, the final meshes, from the iterative solution satisfying my stopping criteria and from the iterative solution satisfying the heuristic stopping criterion, are almost identical. On the other hand, no savings on computation time are found when applying these stopping criteria to the GMRES solver with line Gauss-Seidel preconditioner.

For completeness, my dissertation also includes proofs of the existence of the finite element solutions (both GFEM and SDFEM), the a priori error estimations, the a posteriori error estimations and the convergence of MG and AMG. The deductions of the moving mesh algorithm based on the equidistribution principle and the GMRES algorithm are also shown in my dissertation.

Summary and Future Works:

In summary, my dissertation shows that the multigrid iteration with new stopping criteria is a very efficient solver for the linear system obtained from SDFEM discretization on adaptively refined meshes. The new error-adapted refinement algorithm significantly improves the solution accuracy in layer regions. Even with these improvements, the search for an accurate and efficient solver of the convection-diffusion equation is in the early stages. Many questions remain to be answered. For example, recently, Xu and Zikatanov propose a new edge-averaged finite element discretization scheme called EAFE for the convection-diffusion equation. The discrete matrix from this scheme is an M-matrix. As a result, one may expect to obtain better performance in solving linear systems, obtained from EAFE discretization, than in solving linear systems, obtained from SDFEM discretization. Can this be true? Can one obtain the a posteriori error estimations for other discretization schemes such as EAFE or the discontinuous Galerkin method? Anisotropic meshes are generally used in real applications. Recently, Kunert has developed an a posteriori error estimator for the convection-diffusion equation on some anisotropic meshes. Can such an a posteriori error estimator be used on the error-adapted refined meshes and therefore make the error-adapted refinement process more reliable? How do MG methods and AMG methods perform, under anisotropic meshes and error-adapted refined meshes, for convection dominant flow problems? When a relatively small minimal-resolution of the approximate solution is required on the whole domain, the number of points in coarse meshes can be very large. Grid coarsening becomes an issue when MG solvers are applied to coarse problems. In contrast, AMG is ready to be used on coarse problems in this case. However, can we prove MG or AMG converge for the case in which mesh Peclét number is much greater than one? What stopping criteria of the iterative solvers should be used if the linear systems are obtained from discretization under the anisotropic meshes? In the near future, I would like to gain more insight regarding answers to the above questions. I also would like to take the next step toward to solving the Naiver-Stokes equation. I look forward to facing the new challenges that come along with these goals.