Discontinuous Galerkin Methods for a Class of Nonvariational Problems

We extend the finite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2): 786–801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the “finite element Hessian” as an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et al. (SIAM J. Numer. Anal. 39(5): 1749–1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the finite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting.

Novel Adaptive Hybrid Discontinuous Galerkin Algorithms for Elliptic Problems

In this work, we develop novel adaptive hybrid discontinuous Galerkin algorithms for second-order elliptic problems. For this, two types of reliable and efficient, modulo a data-oscillation term, and fully computable a posteriori error estimators are developed: the first one is a simple residual type error estimator, and the second is a flux reconstruction based error estimator of a guaranteed type for polynomial approximations of any degree by using a simple postprocessing. These estimators can achieve high-order accuracy for both smooth and nonsmooth problems even with high-order approximations. In order to enhance the performance of adaptive algorithms, we introduce 𝐾-means clustering based marking strategy. The choice of marking parameter is crucial in the performance of the existing strategy such as maximum and bulk criteria; however, the optimal choice is not known. The new strategy has no unknown parameter. Several numerical examples are given to illustrate the performance of the new marking strategy along with our estimators.

$ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains

<p style='text-indent:20px;'>In this paper, we establish the <inline-formula><tex-math id="M1">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the <inline-formula><tex-math id="M2">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M3">\begin{document}$ \frac{3}{2}-\varepsilon&lt;p&lt;3+\varepsilon $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M4">\begin{document}$ d\geq3 $\end{document}</tex-math></inline-formula>, and the range for <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula> is sharp. For elliptic systems, we prove that the <inline-formula><tex-math id="M6">\begin{document}$ W^{1,p} $\end{document}</tex-math></inline-formula> estimates hold for <inline-formula><tex-math id="M7">\begin{document}$ \frac{2d}{d+1}-\varepsilon&lt;p&lt;\frac{2d}{d-1}+\varepsilon $\end{document}</tex-math></inline-formula> under the assumption that the Lipschitz constant of the domain is small.</p>