Comparison of different time discretization schemes for solving the Allen–Cahn equation

This article aims to solve the nonlinear Allen–Cahn equation numerically. The diagonally implicit fractional-step θ-(DIFST) scheme is used for the discretization of the time derivative term while the space derivative is discretized by the conforming finite element method. The computational efficiency of the DIFST scheme in terms of CPU time and temporal error estimation is computed and compared with other time discretization schemes. Several test problems are presented to show the effectiveness of the DIFST scheme.

Numerical study of flow across an ellipse and a circle placed in a uniform stream of infinite extent

As an example of an aerodynamics prototypical study, we examined a two-dimensional low Reynolds number flow over obstacles immersed in a stream of infinite extent. The Navier Stokes equation is being discretized by non conforming finite element method approach. The resulting discretized nonlinear algebraic system is being solved by using the fixpoint method and the Newton method and multigrid method for the linear sub-problem employed. The magnitude of the uniform upstream velocity under the study of the problem for Reynolds number in the range 1 < Re < 100 and the angle of attack of the upstream velocity at α = -5; 0; 5 degrees performed. Analysis of the resulting drag and lift forces acting on obstacles with respect to the angle of attack of the upstream velocity and the Reynolds number is made. Moreover, the influence of one obstacle on the resulting drag and lift coefficients of other obstacles determined. The results are being presented in a graphical and vector form.

A priori error estimates for a linearized fracture control problem

A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived.

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time $t=0$ t = 0 . Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as $r\geq (2-\alpha )/\alpha $ r ≥ ( 2 − α ) / α . Numerical results confirm that our analysis is sharp.