Let $u=x^{2}+2y^{2}$. By AM-GM, $u \geq \sqrt{8}xy$, so $xy \leq \frac{u}{\sqrt{8}}$. If we let $xy=ku$ where $k \leq \frac{1}{\sqrt{8}}$, then we have $u(1-k)=8$ and $u(1+k)=x^{2}+xy+2y^{2}$, that is, $u(1+k)=8 \cdot \frac{1+k}{1-k}$. It is not hard to see that the maximum value of this expression occurs at $k=\frac{1}{\sqrt{8}}$, so the maximum value is $8 \cdot \frac{1+\frac{1}{\sqrt{8}}}{1-\frac{1}{\sqrt{8}}}=\frac{72+32 \sqrt{2}}{7}$.