Maximum-norm estimates of an IP-HDG method for parabolic problems on general polyhedral meshes

In this paper we study maximum-norm estimates for the semidiscrete problem of parabolic equations in a convex polygonal/polyhedral domain with an interior penalty hybridized discontinuous Galerkin (IP-HDG) method. For the homogeneous case with zero source/sink the maximum-norm stability of the semigroup defined by the semidiscrete IP-HDG method is proved based on a local energy error estimate, a dyadic decomposition of the space time domain and local energy estimates of Green’s and regularized Green’s functions. For the inhomogeneous case we establish error estimates in L∞((0,T);Lₕ∞( )) and L∞((0,T);Wₕ1,∞( )) norms by maximum-norm error estimates of elliptic problems and energy estimates of regularized Green’s function and its IP-HDG discretization, where Lₕ∞( ) and Wₕ1,∞( ) are mesh-dependent Sobolev spaces defined in Section 2. It is worth noting that the proof of the local energy error estimate, a key component of this paper, has not used the lifting operator that is only available for simplicial meshes. Thus, the results of this paper are applicable for polygonal/polyhedral meshes.