In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, $(p-1)$ functions per edge, and $(p-1)(p-2)/2$ interior functions that span the ${\{x^a{y}^b\}}_{(a+b)\le p}$ space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

在本文中，我们针对一般多边形网格上的二阶椭圆偏微分方程，使用减少空间多边形Bernstein-Bézier函数，给出了对称的内部罚分不连续Galerkin有限元离散化方法。它们形成了一个插值函数（局部多项式函数）的空间，这些插值函数在其插值点处满足Lagrange属性。这可以为基于挑战的渐近扩散极限中的任意多边形上的一阶线性传输方程使用有效的基于DGFEM物理的扩散预处理器。在具有n个顶点和多项式阶数p的多边形元素上，有n个顶点函数，$（p-\mathrm{1个}）$ 每个边的功能，以及 $（p-\mathrm{1个}）（p-\mathrm{2个}）/\mathrm{2个}$ 内部功能跨越 ${\{X^{\mathrm{一种}}{ÿ}^b\}}_{（\mathrm{一种}+b）\le p}$二元单项式的空间。通过均匀，自适应的h，p和hp细化策略，提出了突出提出的方法性能的数值实验。观察到适当的错误率，包括hp适应性，在解决方案中存在奇异点时会产生指数收敛。