An assessment of discretizations for convection-dominated convection-diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented

An assessment of discretizations for convection-dominated convection-diffusion equations

The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented

Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach

In this study, we consider the simulation of subsurface flow and solute transport processes in the stationary limit. In the convection-dominant case, the numerical solution of the transport problem may exhibit non-physical diffusion and under- and overshoots. For an interior penalty discontinuous Galerkin (DG) discretization, we present a $h$-adaptive refinement strategy and, alternatively, a new efficient approach for reducing numerical under- and overshoots using a diffusive $L^2$-projection. Furthermore, we illustrate an efficient way of solving the linear system arising from the DG discretization. In $2$-D and $3$-D examples, we compare the DG-based methods to the streamline diffusion approach with respect to computing time and their ability to resolve steep fronts

On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion-reaction equations

A standard discontinuous Galerkin (DG) finite element method for discretizing steady-state convection-diffusion-reaction equations is known to be stable and to compute sharp layers in the convection-dominated regime, but also to show large spurious oscillations. This paper studies post-processing methods for reducing the spurious oscillations, which replace the DG solution in a vicinity of layers by a constant or linear approximation. Three methods from the literature are considered and several generalizations and modifications are proposed. Numerical studies with the post-processing methods are performed at two-dimensional examples

On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion-reaction equations

A standard discontinuous Galerkin (DG) finite element method for discretizing steady-state convection–diffusion equations is known to be stable and to compute sharp layers in the convection-dominated regime, but also to show large spurious oscillations. This paper studies post-processing methods for reducing the spurious oscillations, which replace the DG solution in a vicinity of layers by a constant or linear approximation. Three methods from the literature are considered and several generalizations and modifications are proposed. Numerical studies with the post-processing methods are performed at two-dimensional examples

A unified analysis of algebraic flux correction schemes for convection–diffusion equations

Recent results on the numerical analysis of algebraic flux correction (AFC) finite element schemes for scalar convection–diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme

A stabilised finite element method for the convection-diffusion-reaction equation in mixed form

This paper is devoted to the approximation of the convection-diffusion-reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems

On slope limiting and deep learning techniques for the numerical solution to convection-dominated convection-diffusion problems

As the first main topic, several slope-limiting techniques from the literature are presented, and various novel methods are proposed. These post-processing techniques aim to automatically detect regions where the discrete solution has unphysical values and approximate the solution locally by a lower degree polynomial. This thesis's first major contribution is that two novel methods can reduce the spurious oscillations significantly and better than the previously known methods while preserving the mass locally, as seen in two benchmark problems with two different diffusion coefficients. The second focus is showing how to incorporate techniques from machine learning into the framework of classical finite element methods. Hence, another significant contribution of this thesis is the construction of a machine learning-based slope limiter. It is trained with data from a lower-order DG method from a particular problem and applied to a higher-order DG method for the same and a different problem. It reduces the oscillations significantly compared to the standard DG method but is slightly worse than the classical limiters. The third main contribution is related to physics-informed neural networks (PINNs) to approximate the solution to the model problem. Various ways to incorporate the Dirichlet boundary data, several loss functionals that are novel in the context of PINNs, and variational PINNs are presented for convection-diffusion-reaction problems. They are tested and compared numerically. The novel loss functionals improve the error compared to the vanilla PINN approach. It is observed that the approximations are free of oscillations and can cope with interior layers but have problems capturing boundary layers

Shock capturing techniques for hp-adaptive finite elements

The aim of this work is to propose an hp-adaptive algorithm for discontinuous Galerkin methods that is capable to detect the discontinuities and sharp layers and avoid the spurious oscillation of the solution around them. In order to control the spurious oscillations, artificial viscosity is used with the particularity that it is only applied around the layers where the solution changes abruptly. To do so, a novel troubled-cell detector has been developed in order to mark the elements around those layers and to impose linear order in them. The detector takes advantage of the evolution of the value of the gradient through the adaptive process.Peer ReviewedPostprint (published version