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Gabriel R. Barrenechea, Volker John, Petr Knobloch
Finite element methods respecting the discrete maximum principle for
convection-diffusion equations, SIAM Review 66, 3 - 88, 2024
Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation. |
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Abhinav Jha, Volker John, Petr Knobloch
Adaptive Grids in the Context of Algebraic Stabilizations for Convection-Diffusion-Reaction Equations, SIAM J. Sci. Comp. 45, No. 4, B564 - B589, 2023
Three algebraically stabilized finite element schemes for discretizing convection-diffusion-reaction equations are studied on adaptively refined grids. These schemes are the algebraic flux correction (AFC) scheme with Kuzmin limiter, the AFC scheme with BJK limiter, and the recently proposed Monotone Upwind-type Algebraically Stabilized (MUAS) method. Both, conforming closure of the refined grids and grids with hanging vertices are considered. A non-standard algorithmic step becomes necessary before these schemes can be applied on grids with hanging vertices. The assessment of the schemes is performed with respect to the satisfaction of the global discrete maximum principle (DMP), the accuracy, e.g., smearing of layers, and the efficiency in solving the corresponding nonlinear problems.
Volker John, Petr Knobloch, Ondřej Pártl
A Numerical Assessment of Finite Element Discretizations for Convection-Diffusion-Reaction Equations Satisfying Discrete Maximum Principles, Comp. Meth. Appl. Math. 23, 969 - 988, 2023
| Numerical studies are presented that investigate finite element methods satisfying discrete maximum principles for convection-diffusion-reaction equations. Two linear methods and several nonlinear schemes, some of them proposed only recently, are included in these studies, which consider a number of two-dimensional examples. The evaluation of the results examines the accuracy of the numerical solutions with respect to quantities of interest, like layer widths, and the efficiency of the simulations.
Volker John, Petr Knobloch
On algebraically stabilized schemes for convection-diffusion-reaction problems, Numerische Mathematik 152, 553 - 585, 2022
| An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection--diffusion--reaction equations. The definition of this scheme contains a new limiter that improves a standard one in such a way that local and global discrete maximum principles are satisfied on arbitrary simplicial meshes.
Volker John, Paul Korsmeier, Petr Knobloch
On the Solvability of the Nonlinear Problems in an Algebraically
Stabilized Finite Element Method for Evolutionary Transport-Dominated Equations, Math. Comp. 90, 595 - 611 , 2021
| The so-called FEM-FCT (finite element method flux-corrected transport) scheme for evolutionary scalar convection-dominated equations leads in each time instant to a nonlinear problem. For sufficiently small time steps, the existence and uniqueness of a solution of these problems is shown. Moreover, the convergence of a semi-smooth Newton's method is studied.
Volker John, Petr Knobloch
Existence of solutions of a finite element flux-corrected-transport scheme, Appl. Math. Lett. 115, Article 106932, 2021
| The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection-diffusion-reaction equations and transport equations.
Abhinav Jha, Volker John
A Study of Solvers for Nonlinear AFC Discretizations of Convection-Diffusion Equations, Computers and Mathematics with Applications 78, 3117 - 3138, 2019
| Nonlinear discretizations are necessary for convection-diffusion equations for obtaining accurate solutions that satisfy the discrete maximum principle. The numerical solution of the arising nonlinear problems is often difficult. This paper presents several approaches for solving the nonlinear problems of algebraic flux correction (AFC) schemes for the Kuzmin limiter and the BJK limiter. Comprehensive numerical studies are performed at examples that model the transport of energy from a body in a flow field in two and three dimensions. It turns out that the most efficient approach, from the point of view of computing times, is a simple fixed point iteration, because the iteration matrix possesses properties that can be exploited by the solvers of the arising linear systems of equations.
Gabriel R. Barrenechea, Volker John, Petr Knobloch, Richard Rankin
A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations,
SeMA Journal 75, 655 - 685, 2018
| 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 scheme. Then, specific versions of the method, this is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.
Gabriel R. Barrenechea, Volker John, Petr Knobloch A linearity preserving algebraic flux correction scheme satisfying the discrete maximum principle on general meshes,
Mathematical Models and Methods in Applied Sciences (M3AS) 27, 525 - 548, 2017
| This work is devoted to the proposal of a new flux limiter that makes the algebraic flux correction finite element scheme linearity and positivity preserving on general simplicial meshes. Minimal assumptions on the limiter are given in order to guarantee the validity of the discrete maximum principle, and then a precise definition of it is proposed and analyzed. Numerical results for convection-diffusion problems confirm the theory.
Gabriel R. Barrenechea, Volker John, Petr Knobloch
Analysis of algebraic flux correction schemes,
SIAM J. Numer. Analysis 54, 2427 - 2451, 2016
| A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection--diffusion--reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.
Gabriel R. Barrenechea, Volker John, Petr Knobloch
Some analytical results for an algebraic flux correction scheme for a steady
convection--diffusion equation in 1D,
IMA J. Numer. Anal. 35, 1729 - 1756, 2015
| Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection--diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method. |
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Derk Frerichs-Mihov, Linus Henning, Volker John
Using deep neural networks for detecting spurious oscillations in discontinuous Galerkin solutions of convection-dominated convection-diffusion equations, J. Sci. Comp. 97, Article 36, 2023
Standard discontinuous Galerkin (DG) finite element solutions to convection-dominated con\-vec\-tion-diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem. | |||
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Derk Frerichs, Volker John
On a technique for reducing spurious
oscillations in DG solutions of convection-diffusion equations, Appl. Math. Lett. 129, Article 107969, 2022
This note studies a generalization of a post-processing technique and a novel method inspired by the same technique which significantly reduce spurious oscillations in discontinuous Galerkin solutions of con\-vec\-tion-diffusion equations in the convection-dominated regime.
Derk Frerichs, Volker John
On Reducing Spurious Oscillations in Discontinuous Galerkin (DG) Methods for Steady-State Convection-Diffusion Equations, J. Comp. Appl. Math. 393, Article 113487, 2021
| 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. |
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Volker John, Petr Knobloch, Ulrich Wilbrandt
A posteriori optimization of
parameters in stabilized methods for convection-diffusion problems - Part
II, J. Comp. Appl. Math. 428, Article 115167, 2023
Extensions of algorithms for computing optimal stabilization parameters in finite element methods for convection-diffusion equations are presented. These extensions reduce the dimension of the control space, in comparison to available methods, and thus address the long computing times of these methods. One method is proposed that considers only relevant mesh cells, another method that uses groups of mesh cells, and the combination of both methods is also studied. The incorporation of these methods within a gradient-based optimization procedure, via solving an adjoint problem, is explained. Numerical studies provide impressions on the gain of efficiency as well as on the loss of accuracy if control spaces with reduced dimensions are utilized.
Volker John, Petr Knobloch, Simona B. Savescu
A posteriori optimization of parameters in stabilized methods for
convection-diffusion problems - Part I
, Comput. Meth. Appl. Mech. Engrg. 200, 2916 - 2929, 2011
| Stabilized finite element methods for convection-dominated problems require the choice of appropriate stabilization parameters. From numerical analysis, often only their asymptotic values are known. This paper presents a general framework for optimizing stabilization parameters with respect to the minimization of a target functional. Exemplarily, this framework is applied to the SUPG finite element method and the minimization of a residual-based error estimator, an error indicator, and a functional including the crosswind derivative of the computed solution. Benefits of the basic approach are demonstrated by means of numerical results. |
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Volker John, Xu Li, Christian Merdon, Hongxing Rui
Inf–sup stabilized Scott–Vogelius pairs on general shape-regular simplicial grids by Raviart–Thomas enrichment,
Mathematical Models and Methods in Applied Sciences 34, 919 - 949, 2024
This paper considers the discretization of the Stokes equations with Scott--Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order $k$ of the Scott--Vogelius velocity space with appropriately chosen and explicitly given Raviart--Thomas bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 42 (2022), 3711--3734], where the case $k=1$ was studied. The proposed method is pressure-robust, with optimally converging $\vecb{H}^1$-conforming velocity and a small $\vecb{H}(\mathrm{div})$-conforming correction rendering the full velocity divergence-free. For $k\ge d$, with $d$ being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart--Thomas enrichment and also all non-constant pressure degrees of freedom can be eliminated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent $\vecb{P}_k \times P_0$ scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.
Volker John, Alexander Linke, Christian Merdon, Michael Neilan, Leo R. Rebholz
On the divergence constraint in mixed finite element methods for incompressible flows,
SIAM Review 59, 492 - 544, 2017
| The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $\bH(\mathrm{div})$-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.
E.W. Jenkins, V. John, A. Linke, L.G. Rebholz
On the parameter choice in grad-div stabilization for
the Stokes equations, Adv. Comput. Math. 40, 491 - 516, 2014
| Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the $H^1(\Omega)$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the $L^2(\Omega)$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the $H^1(\Omega)$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new. |