A New Class of Adaptive Discontinuous Petrov-Galerkin (DPG) Finite Element (FE) Methods with Application to Singularly Perturbed Problems
Adaptive finite elements vary element size h or/and polynomial order p to deliver approximation properties superior to standard discretization methods. The best approximation error may converge even exponentially fast to zero as a function of problem size (CPU time, memory). The adaptive methods are thus a natural candidate for singularly perturbed problems like convection-dominated diffusion, compressible gas dynamics, nearly incompressible materials, elastic deformation of structures with thin-walled components, etc. Depending upon the problem, diffusion constant, Poisson ratio or beam (plate, shell) thickness, define the small parameter. This is the good news. The bad news is that only a small number of variational formulations is stable for adaptive meshes By the stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of the mesh. To this class belong classical elliptic problems (linear and non-linear), and a large class of wave propagation problems whose discretization is based on hp spaces reproducing the classical exact grad-curl-div sequence. Examples include acoustics, Maxwell equations, elastodynamics, poroelasticity and various coupled and multiphysics problems. For singularly perturbed problems, the method should also be robust, i.e. the stability constant should be independent of the perturbation parameter. This is also the dream for wave propagation problems in the frequency domain where the (inverse of) frequency can be identified as the perturbation parameter. In this context, robustness implies a method whose stability properties do not deteriorate with the frequency (method free of pollution (phase) error). We will present a new paradigm for constructing discretization schemes for virtually arbitrary systems of linear PDE's that remain stable for arbitrary hp meshes, extending thus dramatically the applicability of hp approximations. The DPG methods build on two fundamental ideas: - a Petrov-Galerkin method with optimal test functions for which continuous stability automatically implies discrete stability, - a discontinuous Petrov-Galerkin formulation based on the so-called ultra-weak variational hybrid formulation. We will use linear acoustics and convection-dominated diffusion as model problems to present the main concepts and then review a number of other applications for which we have collected some numerical experience including: 1D and 2D convection-dominated diffusion (boundary layers) 1D Burgers and compressible Navier-Stokes equations (shocks) Timoshenko beam and axisymmetric shells (locking, boundary layers) 2D linear elasticity (mixed formulation, singularities) 1D and 2D wave propagation (pollution error control) 2D convection and 2D compressible Euler equations (contact discontinuities and shocks) The presented methodology incorporates the following features: The problem of interest is formulated as a system of first order PDE's in the distributional (weak) form, i.e. all derivatives are moved to test functions. We use the DG setting, i.e. the integration by parts is done over individual elements. As a consequence, the unknowns include not only field variables within elements but also fluxes on interelement boundaries. We do not use the concept of a numerical flux but, instead, treat the fluxes as independent, additional unknowns (a hybrid method). For each trial function corresponding to either field or flux variable, we determine a corresponding optimal test function by solving an auxiliary local problem on one element. The use of optimal test functions guarantees attaining the supremum in the famous inf-sup condition from Babuska-Brezzi theory. The resulting stiffness matrix is always hermitian and positive- definite. In fact, the method can be interpreted as a least-squares applied to a preconditioned version of the problem. By selecting right norms for test functions, we can obtain stability properties uniform not only with respect to discretization parameters but also with respect to the perturbation parameter (diffusion constant, Reynolds number, beam or shell thickness, wave number) In other words, the resulting discretization is robust.
