The virtual element method with interior penalty for the fourth-order singular perturbation problem

We present the virtual element method with interior penalty to solve a fourth-order singular perturbation problem. In order to estimate the nonconformity error, the degrees of freedom on edges are changed to the moments of functions in the interior penalty scheme. To do this, we design a special H-1-type projection that can be uniquely determined by the new degrees of freedom. With the help of the H-1-type projection, the local discrete space is defined by imposing some restrictions on the local space of H-2-conforming VE, such that it has the local H-2 regularity. Then for the numerical method, we derive the error estimates under the condition of enough smooth solution and the uniform error estimates for cases with boundary layers. Finally, we display some numerical examples. From that, we see that the interior penalty method gives a better performance on the convergence than the theoretical predictions.