Hybrid Discontinuous Galerkin methods for elliptic problems based on a Least-Squares variational principle

Resumo

We propose new hybrid finite element methods for elliptic problems based on a Least-Squares varia-
tional principle (LS-h). We devised the LS-h formulation considering local minimization problems in each element

of the mesh, with the objective function composed of Least-Squares residual terms in each element and local in-
terface conditions (i.e., transmission conditions on the mesh skeleton). The LS-h formulation can be rewritten in

terms of independent local problems and a coupled global problem. The former consists of Least-Squares formu-
lations and the latter is written in terms of a Lagrange multiplier – identified as the trace of the primal variable –

imposing the transmission condition on the mesh skeleton. Thus, we obtain the global system by static conden-
sation, reducing considerably the number of unknowns to be solved. For the resulting algebraic system, through

Singular Value Decomposition (SVD) numerical calculations, we estimate the condition number of the LS-h using
the l
2
-norm. We compare the LS-h with classical Hybridizable Discontinuous Galerkin (HDG), showing that LS-h
has similar condition number estimates in spite of the different block structure in its resulting system. Furthermore,
we performed numerical experiments using the method of manufactured solutions to show that LS-h has optimal
convergence rates – in terms of l
2
-norm – for both primal and flux variables.