Organizer: Takuya TSUCHIYA
Contact: tsuchiya -at- math -dot- sci -dot- ehime-u -dot- ac -dot- jp
We plan a workshop on numerical analysis for partial differential equations, and invite young researchers from the Hong Kong Polytechnic University and Nagoya University.
13:30 -- 14:10
Takuya TSUCHIYA (Ehime University)
Error analysis of Crouzeix-Raviart and Raviart-Thomas finite element methods
We consider the error analysis of Crouzeix-Raviart and Raviart-Thomas finite element methods on triangulations that do not satisfy the regularity condition. Similar to the case of piecewise $k$th order Lagrange finite element method, the error is bounded in terms of the diameter and circumradius of triangles in triangulations. Numerical examples will be given.
14:20 -- 15:00
Zhonghua QIAO (The Hong Kong Polytechnic University)
Semi-implicit methods for phase field equations
Recent results in the literature provide computational evidence that the stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth order nonlinear diffusion. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the derivative of the nonlinear potential function satisfies a Lipschitz-type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work, we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of the stabilization term depends on the initial energy and the perturbation parameter but is independent of the time step.
15:10 -- 15:50
Tomoya KENMOCHI (Nagoya University)
Maximum norm error estimates for the finite element approximation of parabolic problems on smooth domains
In this talk, we consider the finite element method for parabolic problems on smooth and generally non-convex domains. In order to implement FEM on such domains, we approximate the domain by polygonal domains, and then we can consider FEM on the approximate domains. We will present a maximum norm error estimate and show that the error of domain approximation is dominant for higher order elements. The smoothing property and the maximal regularity for discrete elliptic operators will be presented as corollaries.
16:00 -- 16:40
Zhi ZHOU (The Hong Kong Polytechnic University)
High-order time-stepping schemes for time-fractional diffusion equations
The time-fractional diffusion, which has received much attention in recent years, describes a diffusion process in which the mean square displacement of a particle grows slower (sub-diffusion) than that in the normal diffusion process. The solution of the fractional diffusion often exhibits a singular layer, provided that the source data is not compatible with the initial data, which makes the numerical treatment and analysis challenging. We develop a systematic strategy to the starting k-1 steps in order to restore the desired kth-order convergence rate of the k-step BDF convolution quadrature for the time-fractional equations. The desired kth-order convergence rate can be achieved even if the solution is non-smooth.