• Abstract

Novelty of this work is the development of a finite element method using discontinuous $P_k$ element, which has two-order higher convergence rate than the optimal order. The method is used to solve a one-dimensional second order elliptic problem. A totally new approach is developed for error analysis. Superconvergence of order two for the CDG finite element solution is obtained. The $P_k$ solution is lifted to an optimal order $P_{k+2}$ solution elementwise. The numerical results confirm the theory.

• Abstract

The stability and uniqueness of the solutions of time-fractional telegraph equations arising in the transmission line modeling are proved. The corresponding initial-boundary problems are then solved by a finite difference scheme. It is shown that the scheme is unconditionally stable and convergent. Computational efficiency of the method can be enhanced by transforming it into two finite volume schemes for solving two uncoupled time-fractional convection equations. Numerical experiments validate the theoretical results and show the efficiency of this approach even for the problems the solutions which are not smooth at the initial moment.

• Abstract

The forager-exploiter model with gradient-dependent flux-limitation$$u_t=\Delta u-\chi \nabla \cdot\left(uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}}\right)\nabla w,$$ $$v_t=\Delta v-\xi \nabla \cdot\left(vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}}\right)\nabla u,$$ $$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$ is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N$, $N ≥ 2.$ It is shown that if $α >(N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0$, $v_0$, $w_0 ∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w)\in (C^0(\overline{\Omega}\times[0,∞))\cap C^{2,1}(\overline{\Omega}\times(0,∞)))^3.$