A BDDC Algorithm for Weak Galerkin Discretizations

The weak Galerkin (WG) methods are a class of nonconforming finite element methods, which were first introduced for a second order elliptic problem in Wang and Ye (2014). The idea of the WG is to introduce weak functions and their weak derivatives as distributions, which can be approximated by polynomials of different degrees. For second elliptic problems, weak functions have the form of v = { v ₀, v _b }, where v ₀ is defined inside each element and v _b is defined on the boundary of the element. v ₀ and v _b can both be approximated by polynomials. The gradient operator is approximated by a weak gradient operator, which is further approximated by polynomials. These weakly defined functions and derivatives make the WG methods highly flexible and these WG methods have been extended to different applications such as Darcy in Lin et al. (2014), Stokes in Wang and Ye (2016), bi-harmonic in Mu et al. (2014), Maxwell in Mu et al. (2015c), Helmholtz in Mu et al. (2015b), and Brinkman equations in Mu et al. (2014). In Mu et al. (2015a), the optimal order of polynomial spaces is studied to minimize the number of degrees of freedom in the computation.