A penalty-free $hp$-version mixed discontinuous Galerkin method for the biharmonic equation

This paper focused on an $hp$-version mixed discontinuous Galerkin method without penalty terms for the biharmonic equation with Navier boundary conditions. By introducing the auxiliary variable $v=u_{xx}$, we reduced the fourth-order problem to a second-order system and derived its penalty-free variational formulation. The analysis replaces the standard coercivity condition with a polynomial-degree-dependent inf-sup condition for the bilinear form $B(\cdot ,\cdot )$. While $h$-convergence rates for both $u_h$ and $v_h$ were optimal, the $p$-convergence exhibited contrasting behavior: suboptimal in $L^2$-norm but optimal in the energy norm, regardless of the $p^2$ scaling in the inf-sup condition. Numerical results revealed that $p$-convergence order-doubling for boundary-aligned singularities significantly enhanced the efficacy for singular solutions. Furthermore, the method was shown to extend to nonlinear biharmonic equations, while the treatment of Dirichlet boundary conditions necessitated the introduction of penalty terms.