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Recent years have witnessed the growing presence of artificial intelligence in language learning apps. Against this backdrop, this technology review provides an overview of the affordances of Speeko and discusses its potential in developing English-as-a-foreign-language students' public speaking skills. [ABSTRACT FROM AUTHOR]

As is well known, piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC) are used to approximate the weakly singular integrals $$\begin{equation*}I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}\textrm{d}y, \quad x \in (a,b),\quad 0< \gamma <1,\end{equation*}$$ which have local truncation errors |$\mathcal{O} (h^2)$| and |$\mathcal{O} (h^{4-\gamma })$|⁠ , respectively. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e. |$\lambda u(x)- I(a,b,x) =f(x)$|⁠ , |$\lambda \neq 0$|⁠ , the global convergence rates are also |$\mathcal{O} (h^2)$| and |$\mathcal{O} (h^{4-\gamma })$| by PLC and PQC in Atkinson (2009, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press). In this work we study the following nonlocal problems, which are similar to the above Fredholm integral equations: $$\begin{equation*}\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}\textrm{d}y =f(x), \quad x \in (a,b),\quad 0< \gamma <1. \end{equation*}$$ In the first part of this paper we prove that the weakly singular integrals |$I(a,b,x)$| have optimal local truncation error |$\mathcal{O}(h^4\eta _i^{-\gamma })$| by PQC, where |$\eta _i=\min \left \{x_i-a,b-x_i\right \}$| and |$x_i$| coincides with an element junction point. Then the sharp global convergence orders |$\mathcal{O}\left (h\right)$| and |$\mathcal{O} (h^3)$| by PLC and PQC, respectively, are established for nonlocal problems. Finally, numerical experiments are shown to illustrate the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements. As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, a priori error estimates in energy and norms are derived rigorously, and numerical results confirm the effectiveness of the method. [ABSTRACT FROM AUTHOR]

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. These 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 are presented to verify the convergence orders for two finite elements with discontinuous and continuous approximations for the total stress. [ABSTRACT FROM AUTHOR]