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In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the time-dependent Oseen equation. The generalized weak Galerkin method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical scheme are developed and analyzed for their convergence, stability, and error estimates. A generalized {\em{inf-sup}} condition is developed to assist the convergence analysis. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples., Comment: 21 pages, 5 Tables

In this work, semi-discrete and fully-discrete error estimates are derived for the Biot's consolidation model described using a three-field finite element formulation. The fields include displacements, total stress and pressure. The model is implemented using a backward Euler discretization in time for the fully-discrete scheme and validated for benchmark examples. Computational experiments presented verifies the convergence orders for the lowest order finite elements with discontinuous and continuous finite element appropriation for the total stress., Comment: 18 pages, 2 figures

As is well known, using piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC), respectively, to approximate the weakly singular integral $$I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}dy, \quad x \in (a,b) ,\quad 0< \gamma <1,$$ have the local truncation error $\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^{4-\gamma}\right)$. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\lambda u(x)- I(a,b,x) =f(x)$ with $ \lambda \neq 0$, also have global convergence rate $\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^{4-\gamma}\right)$ in [Atkinson and Han, Theoretical Numerical Analysis, Springer, 2009]. Formally, following nonlocal models can be viewed as Fredholm weakly singular integral equations $$\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}dy =f(x), \quad x \in (a,b) ,\quad 0< \gamma <1.$$ However, there are still some significant differences for the models in these two fields. In the first part of this paper we prove that the weakly singular integral by PQC have an optimal local truncation error $\mathcal{O}\left(h^4\eta_i^{-\gamma}\right)$, where $\eta_i=\min\left\{x_i-a,b-x_i\right\}$ and $x_i$ coincides with an element junction point. Then a sharp global convergence estimate with $\mathcal{O}\left(h\right)$ and $\mathcal{O}\left(h^3\right)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, the numerical experiments including two-dimensional case are given to illustrate the effectiveness of the presented method., Comment: 24pages, 2 figures

We introduce a new weak Galerkin finite element method whose weak functions on interior neighboring edges are double-valued for parabolic problems. Based on $(P_k(T), P_{k}(e), RT_k(T))$ element, a fully discrete approach is formulated with implicit $\theta$-schemes in time for $\frac{1}{2}\leq\theta\leq 1$, which include first-order backward Euler and second-order Crank-Nicolson schemes. Moreover, the optimal convergence rates in the $L^2$ and energy norms are derived. Numerical example is given to verify the theory.

In this article a theoretical framework for the Galerkin finite element approximation to the time-dependent Riesz tempered fractional problem is provided without the fractional regularity assumption. Because the time-dependent problems should become easier to solve as the time step $\tau \rightarrow 0$, which correspond to the mass matrix dominant [R. E. Bank and T. Dupont, {\em Math. Comp.}, 153 (1981), pp. 35--51]. Based on the introduced and analysis of the fractional $\tau$-norm, the uniform convergence estimates of the V-cycle multigrid method with the time-dependent fractional problem is strictly proved, which means that the convergence rate of the V-cycle MGM is independent of the mesh size $h$ and the time step $\tau$. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform method, where $N$ is the number of the grid points. To the best of our knowledge, this is the first proof for the convergence rate of the V-cycle multigrid finite element method with $\tau\rightarrow 0$., Comment: 23 pages. 0 figures