Many problems satisfy some sort of maximum principle (MP) or positivity at the continuous level. For steady problems, the MP implies that the solution at the interior is bounded by the solution at the boundary. However, these properties might not be readily satisfied at the discrete level. The violation of the MP by the discrete solution may lead to nonphysical results or even disable nonlinear solvers. Therefore, it is essential to provide numerical schemes that inherit such monotonicity properties, namely discrete maximum principle (DMP) preserving schemes.
Although many DMP-preserving methods for explicit time integration are currently available, monotonic schemes for steady or transient problems with implicit time integration are scarce and not so well developed. Implicit time integration is highly desired for problems with fast time scales without engineering/scientific interest. Nevertheless, several improvements have been developed recently to provide accurate DMP-preserving schemes for implicit time integration [1,2,3]. In particular, in  a stabilized continuous Galerkin (cG) finite element (FE) method is supplemented with an artificial diffusion operator to yield a scheme that is DMP-preserving for strictly acute meshes and piecewise continuous FE. In  the scheme is generalized for arbitrary meshes and a novel differentiable artificial diffusion operator is presented. This last development allows us to dramatically improve the nonlinear convergence of the scheme. Finally in  we extend the previous results to the framework of piecewise linear discontinuous Galerkin FE method.
However, all the schemes previously cited need a piecewise linear spatial discretization and are only unconditionally DMP-preserving for first order time integration. Currently, our work focuses on the extension of the previous works to high order discretizations both in space and time.
This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.
Many systems of partial differential equations governing physical phenomena have some underlying structure, e.g., positivity or some maximum principle. However, to inherit at the discrete level such structure is not an obvious task. We are particularly interested in finite element schemes that preserve at the discrete level maximum principles, especially for steady problems and transient problem with implicit time integration. In order to attain such objective, we consider nonlinear stabilization techniques based on a judiciously chosen artificial viscosity. A complete numerical analysis shows that the resulting schemes are discrete maximum princple (DMP) preserving and Lipschitz continuous.
Unfortunately, these additional terms come with a price. The artificial viscosity term is based on a shock capturing detector, which is highly nonlinear and non-differentiable, make extremely hard the nonlinear convergence, and drastically increase the computational cost with respect to nonstabilized formulations. In order to make these algorithms more applicable to real applications, we propose smoothing techniques that lead to differentiable nonlinear viscosity terms, which combined with Newton’s method, allows us to clearly improve nonlinear convergence.
The framework has been originally developed for linear and continuous finite element spaces on accute meshes , and recently extended to discontinuous Galerkin methods  and arbitrary meshes [3, 4]. Since the schemes rely on the convex hull, they loose the DMP property at the discrete level. Finally, in order to go to high order, we will explore the use of B-spline based discretizations.
The convection-diffusion problem is one of the many that might satisfy a Maximum Principle (MP). The MP for steady problems states that the solution in the whole domain is bounded by the solution at the boundary. Being able to satisfy this principle at the discrete level, namely the Discrete Maximum Principle (DMP) is of particular importance. Otherwise, the resulting solutions can lead to nonphysical results or even spoil nonlinear solvers. Although the literature for DMP-preserving methods based on explicit time integration combined with finite volumes or discontinuous Galerkin (dG) is vast and the methods are well known. In the case of continuous Galerkin (cG) based on implicit time integration this is not the case. For problems with fast time scales without engineering interest or when using meshes with highly refined regions implicit time integration becomes a must.
We are interested in developing DMP-preserving finite element schemes for steady problems and transient ones based on implicit time integration. In particular, we consider nonlinear stabilization techniques based on artificial diffusion complemented with a shock detector to minimize the amount of diffusion added and preserve optimal convergence. Based on this approach we have developed a method for cG  and recently extend it to dG . Both methods proposed therein are based on four key ingredients:
1. The first ingredient is the definition of the shock detector that only activates the nonlinear diffusion around shocks/discontinuities. The use of such schemes was proposed in  for 1D problems and extended to multiple dimensions in . Further, the shock detector leads to so-called linearity preserving methods, i.e., the artificial diffusion vanishes for first order polynomials. This property is related to high-order convergence on smooth regions .
2. The second ingredient is the amount of diffusion to be introduced on shocks, which is the amount of diffusion introduced in a first order linear scheme. In this sense, we consider flux-corrected transport techniques .
3. The third ingredient is the form of the discrete viscous operator. In order to keep the DMP on arbitrary meshes we use graph-theoretic, instead of PDE-based, operators for the artificial diffusion terms. This was proposed by Guermond and coworkers  for linear conservation laws.
4. The fourth ingredient is the perturbation of the mass matrix in order to obtain a LED scheme. We use selective mass lumping based on the shock detector in a similar fashion as in .
Unfortunately, these additional terms come with a price. The artificial viscosity term is highly nonlinear and non-differentiable. Which makes the nonlinear convergence extremely hard and drastically increases the computational cost with respect to non-stabilized formulations. In order to make these algorithms more applicable to real applications, we propose the regularization of nondifferentiable terms, which combined with Newton’s method, allow us to clearly improve nonlinear convergence.
Currently our work focuses on the extension to high order discretizations. The above schemes rely on the convex hull, hence they loose the DMP property at the discrete level for high order Lagrangian finite elements. In order to go to high order, we explore the use of discretizations that are a convex hull even for high order, e.g. B-spline based discretizations.
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton’s method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers’ equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a FE space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity).
Hyperbolic problems can satisfy maximum principles or positivity preservation. In this work, we focus on the development of fully implicit finite ellement methods with nonlinear stabilization based on artificial difusion techniques that keep at the discrete level these interesting properties. The nonlinear viscosity is equal to a nonlinear shock detector times a linear viscosity. The idea is to activate the full linear viscosity on discontinuities/shocks, and switch it off in smooth regions. The shock detector must be such that it takes value 1 on local
extrema (to satisfy discrete maximum principles), and switch off for linear functions, i.e., linearity-preserving (for accuracy purposes). In this sense, we work with shock detectors in the line of . Another ingredient is the expression of the Laplacian term. Following , we consider a graph-Laplacian term. Finally, the linear viscosity is edgebased and follows the ideas in flux-corrected transport methods ; we consider the minimum amount of viscosity needed to prove monotonicity properties. The resulting scheme satisfy discrete maximum principles and positivity, and is local variation diminishing. Further, following the ideas in  we can prove Lipschitz continuity. However, it is not enough to end up with a useful numerical method, since the nonlinear convergence of the resulting algorithms is extremely complicated. In this sense, we have developed a smooth version of the framework, in which we can prove that the resulting nonlinear stabilization term is C2-continuous. The resulting schemes
can still keep the monotonicity properties, but are much cheaper (due to a much better nonlinear convergence). We have considered the Newton method with line search and Anderson acceleration techniques. For a large smoothing parameter, the method is cheaper and more dissipative, whereas for a zero smoothing parameter, we recover the original scheme.