On the relationship of various discontinuous finite element methods for second-order elliptic equations (Q2752288)

The author introduces a family of discontinuous finite element methods for fairly general second-order elliptic equations on the bounded domain \(\Omega \subset \mathbb R^n\), \(n=2\) or 3 with boundary \(\partial \Omega= \overline {\Gamma}_1\cup \overline{\Gamma}_2\), \(\Gamma_1\cap\Gamma_2=\emptyset\): NEWLINE\[NEWLINE -\nabla \cdot (a(\nabla u+bu))+du=f \;\text{in} \Omega,\;u=g_D \;\text{on} \Gamma_1, \;a(\nabla u+bu) \cdot v=g_N \;\text{on} \Gamma_2,NEWLINE\]NEWLINE where \(a(x)\) is a positive semi-definite, bounded, symmetric tensor, \(b(x)\) is a bounded vector, \(d(x)\geq 0\) is bounded, \(f(x)\in L^2(\Omega)\), \(g_D(x)\in H^{1/2}(\Gamma_1)\), \(g_N(x)\in H^{-1/2}(\Gamma_2)\) and \(v\) is the outer unit normal to the domain. Discontinuous methods in mixed form are developed and their stability and convergence are established. NEWLINENEWLINENEWLINEThese methods are also written in a nonmixed formulation by incorporating some projection operators into appropriate discontinuous finite element spaces, and the author discusses their relationships to all existing methods. The proof of an error estimate is presented. The author shows that when discontinuous finite element methods are defined in mixed form, they not only preserve good features of these methods, they also have some advantages over classical Galerkin discontinuous methods such as they are more stable in this form.