\(hp\)-dGFEM for second-order elliptic problems in polyhedra I: Stability on geometric meshes (Q2845605)

The authors introduce and analyze \(hp\)-version discontinuous Galerkin (DG) finite element methods for the numerical approximation of linear second-order elliptic boundary value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, the authors consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. Some interior penalty \(hp\)-DG methods are designed and proved that they are well-defined for problems with singular solutions and stable under the proposed \(hp\)-refinements. Some abstract error bounds are established that will allow to prove exponential rates of convergence in a second part of this work.NEWLINENEWLINEFor Part II, see [the author, SIAM J. Numer. Anal. 51, No. 4, 2005--2035 (2013; Zbl 1457.65215)].