Discontinuous PetrovGalerkin method based on the optimal test space norm for steady transport problems in one space dimension
We revisit the finite element analysis of convection-dominated flow problems within the recently developed Discontinuous PetrovGalerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the optimal test space norm. This makes the DPG method not only stable but also robust, that is, uniformly stable with respect to the Pclet number in the current application. We employ discontinuous piecewise Bernstein polynomials as trial functions and construct a subgrid discretization that accounts for the singular perturbation character of the problem to resolve the corresponding optimal test functions. We also show that a smooth B-spline basis has certain computational advantages in the subgrid discretization. The overall effectiveness of the algorithm is demonstrated on two problems for the linear advectiondiffusion equation.
discontinuous petrovgalerkin convectiondiffusion finite element method unconditional stability