An Adaptive Global-Local Generalized Finite Element Method for Transient Partial Differential Equations
Advisors: Prof. Armand Duarte and Prof. Albert Valocchi
Zoom link https://illinois.zoom.us/j/84288506945?pwd=dE9oZVFjV2xDbWdBOWxMbGwvS21Zdz09
This dissertation presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with spatial features smaller than the coarse element size. It allows global and local problems to independently choose their own time integrators. One benefit is that it is feasible to use an explicit time integration method for the global problem to solve advection-dominated problems, whereas the fine-scale reference FEM must choose an implicit one due to the fine mesh size. A potential future research direction is to address stiff problems that possess different time scales with different time integrators. Compared to currently available GFEMgl, the proposed method solves transient instead of steady-state PDEs, possibly with different time integrators among the global and local problems. The proposed approach can solve general transient problems, and it is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are analyzed. Compared with direct analysis by fine-scale FEM, numerical results show that the proposed method has the following properties: 1) the accuracy closely matches direct analysis result with a fine mesh; 2) the critical time step size is loosened; 3) optimal convergence rate is achieved; 4) the fine-scale solution can also be retrieved. To improve computation efficiency, an ad-hoc adaptive approach where local problems can be turned on if their solutions are not helpful for the current global time step motivates the fully automated adaptive approach as follows.
A fully automated adaptive algorithm for the Generalized Finite Element Method with global-local enrichment
(GFEMgl) for transient multiscale PDEs is developed. At each time step, the adaptive algorithm detects a subset of global nodes with trivial enrichments, which are exactly or close to linearly dependent from the underlying coarse FEM basis, and then removes them from the global system. It is based on the calculation of the ratio between the largest and smallest singular values of small sub-matrices extracted from the global system of equations which introduces little overhead over the non-adaptive GFEMgl for transient PDEs. Compared to existing adaptive multiscale approaches, where either an a-posterior error estimate, a change in physical quantities, or a local problem residual is calculated, the proposed approach provides an innovative framework based on singular values, and it is more efficient since the calculation does not rely on local problem solutions.