A new class of discontinuous PETROV-GALERKIN (DPG) finite element (FE) methods with application to convection-dominated diffusion problems

The hp-adaptive finite elements combine elements of varying size h and polynomial order p to deliver approximation properties superior
to any other discretization methods. The best approximation error converges exponentially fast to zero as a function of number of degrees-of-freedom. The hp methods are thus a natural candidate for singularly perturbed problems experiencing internal or boundary layers like in compressible gas dynamics.

This is the good news. The bad news is that only a small number of variational formulations is stable for hp-discretizations.
By the hp-stability we mean a situation where the discretization error can be bounded by the best approximation error times a constant that is independent of both h and p. 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-divsequence. Examples include acoustics, Maxwell, elastodynamics,poroelasticity and various coupled and multiphysics problems.

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. For a start, we focus on a challenging model problem - convection dominated diffusion.

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.

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 local problems for determining optimal test functions are solved approximately with an enhanced approximation (a locally
hp-refined mesh).

By selecting right norms for test functions, we can obtain amazing stability properties uniform not only with respect to discretization
parameters but also with respect to the diffusion constant (the resulting discretization is robust).

The presentation will consist of three parts.

In the first part, we will present a general abstract variational
framework for the optimal test functions.

The second part will focus on the analysis of 1D convection-dominated diffusion
problem. We will present a 1D stability analysis.

The third part will focus on adaptivity. We will present numerous numerical examples for 1D and 2D confusion problems. We have been able to solve in a fully automatic mode problems with diffusion constant eps = 10^{-11} in 1D
and eps = 10^{-7} in 2D using hp-adaptivity.

The presentations will be accompanied with live demonstrations of 1D and 2D codes.