On convergence of the <inline-formula><tex-math id="M1">$ C^0 $</tex-math></inline-formula> interior penalty method for <inline-formula><tex-math id="M2">$ m $</tex-math></inline-formula>th Laplace equa

We study the $ C^0 $ interior penalty method for the $ m $th Laplace equation on a bounded Lipschitz polyhedral domain in $ \mathbb{R}^d $, showing, without extra regularity assumption, that the numerical solution converges to the exact solution both under discrete $ H^l $-norms where $ 2\le l\le m-1 $ and in $ H^1_0(\Omega) $. The strategy we applied consists of three steps: carrying a compactness argument to enumerate all limit points of $ u_h $, proving each of them identical to the unique solution to the PDE, and deducing the convergence.