Convergence Analysis of a Symmetric Dual-Wind Discontinuous Galerkin Method

Abstract

A new symmetric discontinuous Galerkin method for second order elliptic problems is analyzed. We show that the numerical method is stable for any positive penalty parameter and converges with optimal order provided the exact solution is sufficiently regular. These results are also shown to hold for some non-positive penalty parameters. Numerical experiments are presented that support the theoretical results.

Appendix: An Auxiliary Result

Lemma 8

Let $T$ be a $d$-dimensional simplex, and let $e\subset \partial T$ be an arbitrary $(d-1)$-dimensional sub-simplex of $T$. Then any $v_h\in \mathbb P _r(T)$ is uniquely determined by the following values

$$\begin{aligned}&\int \limits _T v_h w_h\, dx\qquad \forall w_h\in \mathbb P _{r-1}(T),\end{aligned}$$

(49a)

$$\begin{aligned}&\int \limits _e v_h q_h\, ds\qquad \forall q_h\in \mathbb P _r(e). \end{aligned}$$

(49b)

Proof

Since $\dim \mathbb P _{r-1}(T) +\dim \mathbb P _r(e) = \left( \begin{array}{c}{d+r-1}\\ {d}\end{array}\right) +\left( \begin{array}{c}{d+r-1}\\ {d-1}\end{array}\right) = \left( \begin{array}{c}{d+r}\\ {d}\end{array}\right) = \mathbb P _r(T)$, it suffices to show that if $v_h$ vanishes at the values (49), then $v_h$ is identically zero.

If $v_h$ vanishes at the values listed in (49b), then we easily conclude that $v_h|_e = 0$. Therefore, we may write $v_h = \lambda _e r_h$ for some $r_h\in \mathbb P _{r-1}(T)$, where $\lambda _e$ is the barycentric coordinate satisfying $\lambda _e|_e = 0$. Since $\lambda _e>0$ in $T$, we conclude from (49a) that $v_h\equiv 0$. $\square$