Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Stabilized discontinuous Galerkin methods for solving hyperbolic conservation laws on grids with embedded objects

This thesis covers a novel penalty stabilization for solving hyperbolic conservation laws using discontinuous Galerkin methods on grids with embedded objects. We consider cut cell grids, that are constructed by cutting the given object out of a Cartesian background grid. The resulting cut cells require special treatments, e.g., adding stabilization terms. In the context of hyperbolic conservation laws, one has to overcome the small cell problem: standard explicit time stepping becomes unstable on small cut cells when the time step is selected based on larger background cells. This work will present the Domain of Dependence (DoD) stabilization in one and two dimensions. By transferring additional information between the small cut cell and its neighbors, the DoD stabilization restores the correct domains of dependence in the neighborhood of the cut cell. The stabilization is added as penalty terms to the semi-discrete scheme. When combined with a standard explicit time-stepping scheme, the stabilized scheme remains stable for a time-step length based on the Cartesian background cells. Thus, the small cell problem is solved. In the first part of this work, we will consider one-dimensional hyperbolic conservation laws. We will start by explaining the ideas of the stabilization for linear scalar problems before moving to non-linear problems and systems of hyperbolic conservation laws. For scalar problems, we will show that the scheme ensures monotonicity when using its first-order version. Further, we will present an L2 stability result. We will conclude this part with numerical results that confirm stability and good accuracy. These numerical results indicate that for both, linear and non-linear problems, the convergence order in various norms for smooth tests is p+1 when using polynomials of degree p. In the second part, we will present first ideas for extending the DoD stabilization to two dimensions. We will consider different simplified model problems that occur when using two-dimensional cut cell meshes. An essential step for the extension to two dimensions will be the construction of weighting factors that indicate how we couple the multiple cut cell neighbors with each other. The monotonicity and L2 stability of the stabilized system will be confirmed by transferring the ideas of the proof from one to two dimensions. We will conclude by presenting numerical results for advection along a ramp, demonstrating convergence orders of p+1/2 to p+1 for polynomials of degree p. Additionally, we present preliminary results for the two-dimensional Burgers and Euler equations on model meshes

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L 2 -energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples