# Jean-Luc Guermond's research while affiliated with Texas A&M University and other places

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## Publications (329)

We study swirling electrovortex flows in a cylinder filled with GaInSn metal using axisymmetric and large-scale three-dimensional numerical simulations. In our set-up electrical currents enter and exit the cell symmetrically through wires and the result is a von Kármán-like flow. Three inductionless and an inductive flow regimes are identified. Sca...

This paper is concerned with the approximation of the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. The proposed approximation technique is robust, formally second-order accurate in space, invariant-domain preserving, and works for every equation of state, tabulated or analytic, provided the pressure is...

In this paper, we introduce a numerical method for approximating the dispersive Serre–Green–Naghdi equations with topography using continuous finite elements. The method is an extension of the hyperbolic relaxation technique introduced in Guermond et al. (J Comput Phys 450:110809, 2022). It is explicit, second-order accurate in space, third-order a...

We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of fractional-order Sobolev spaces. By assuming additionally that the target field has a curl or divergence property, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using...

An invariant domain preserving arbitrary Lagrangian-Eulerian method for solving non-linear hyperbolic systems is developed. The numerical scheme is explicit in time and the approximation in space is done with continuous finite elements. The method is made invariant domain preserving for the Euler equations using convex limiting and is tested on var...

We study swirling electrovortex flows in a cylinder filled with GaInSn metal using axisym-metric and large scale three-dimensional numerical simulations. In our setup , electrical currents enter and exit the cell symmetrically through wires and the result is a von Kármán-like flow. Three induction-less and a new inductive flow regimes are identifie...

The objective of this paper is to propose a hyperbolic relaxation technique for the dispersive Serre–Green–Naghdi equations (also known as the fully non-linear Boussinesq equations) with full topography effects introduced in [14] and [24]. This is done by revisiting a similar relaxation technique introduced in [17] with partial topography effects....

This paper describes in detail the implementation of a finite element technique for solving the compressible Navier–Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is second-order accurate in time and space. Robustness here means that the method is proved to be invariant domain...

In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal deriv...

This paper describes in detail the implementation of a finite element technique for solving the compressible Navier-Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is second-order accurate in time and space. Robustness here means that the method is proved to be invariant domain...

Using direct numerical simulations, we show that swirling electrovortex flows significantly enhance the mixing of the bottom layer alloy in liquid metal batteries during discharge. By studying the flow in various parameter regimes, we identify and explain a novel scaling law for the intensity of these swirling electrovortex flows. Using this scalin...

Combining theoretical arguments and numerical simulations, we demonstrate that the metal pad roll instability can occur in a centimeter-scale setup with reasonable values of the magnetic field and electrical current and with metal pairs that are liquid at room temperature. We investigate two fluid pairs: gallium with mercury (immiscible pair) or ga...

The main objective of this chapter is to present a technique to treat Dirichlet boundary conditions in a natural way using a penalty method. This technique is powerful and has many extensions. In particular, the idea is reused in the next chapter for discontinuous Galerkin methods. Another objective of this chapter is to illustrate again the abstra...

In this chapter, we investigate two additional topics on the approximation of Maxwell’s equations. First, we study the use of a boundary penalty method inspired by Nitsche’s method for elliptic PDEs to enforce the boundary condition on the tangential component. We combine this method with edge (Nédélec) finite elements and with the all-purpose H1-c...

The goal of this chapter is to study the approximation of an elliptic model problem by the discontinuous Galerkin (dG) method. The distinctive feature of dG methods is that the trial and the test spaces are broken finite element spaces. Inspired by the boundary penalty method from the previous chapter, dG formulations are obtained by adding a consi...

In this chapter, we continue the study initiated in the previous chapter. Now that we have in hand our key tool to give a proper meaning to the normal derivative of the exact solution at the mesh faces, we perform the error analysis when the model problem is approximated by one of the nonconforming methods introduced previously, i.e., Crouzeix–Ravi...

Implementing the finite element method requires evaluating the entries of the stiffness matrix and the right-hand side vector, which in turn requires computing integrals over the mesh cells. In practice, these integrals must often be evaluated approximately by means of quadratures. In this chapter, we review multidimensional quadratures that are fr...

The starting point of this chapter is the model problem derived in the previous chapter. Our goal is to specify conditions under which this problem is well-posed. Two important results are presented: the Lax–Milgram lemma and the more fundamental Banach–Nečas–Babuška theorem. The former provides a sufficient condition for well-posedness, whereas th...

This chapter addresses topics related to the approximation of Darcy’s equations using either mixed or H1-conforming finite elements. Mixed finite elements provide a conforming approximation of the dual variable, but the connection to the gradient of the primal variable is enforced weakly. We show here how this connection can be made explicit using...

As in the previous chapters, we want to approximate the Poisson model problem, but this time we use the hybrid high-order (HHO) method. In this method, the discrete solution is composed of a pair: a face component that approximates the trace of the solution on the mesh faces and a cell component that approximates the solution in the mesh cells. The...

The analysis of the previous chapter requires a coercivity property in H(curl). We have also seen that a compactness property needs to be established to deduce an improved L2-error estimate by the Aubin-Nitsche duality argument. We show in this chapter that robust coercivity and compactness can be achieved by a weak control on the divergence of the...

The goal of the present chapter is to identify necessary and sufficient conditions for the well-posedness of a model problem that serves as a prototype for PDEs in mixed form. We consider a setting in Banach spaces and then in Hilbert spaces. The connection with the BNB theorem is highlighted.

The goal of this chapter is to analyze the approximation of second-order elliptic PDEs using H1-conforming finite elements. We focus on homogeneous Dirichlet boundary conditions for simplicity. The well-posedness of the discrete problem follows from the Lax–Milgram lemma and the error estimate in the H1-norm from Céa’s lemma. We also introduce a du...

The goal of this chapter is to derive an a posteriori error estimate for second-order elliptic PDEs approximated by H1-conforming finite elements. Such an estimate is an upper bound on the approximation error that can be computed by using only the discrete solution and the problem data. It can serve the twofold purpose of judging the quality of the...

The present chapter is concerned with the linear elasticity equations where the main tool to establish coercivity is Korn’s inequality. We consider H1-conforming and nonconforming approximations, and we address the robustness of the approximation in the incompressible limit.

A matrix is said to be sparse if the number of its nonzero entries is significantly smaller than the total number of its entries. The stiffness matrix is generally sparse as a consequence of the global shape functions having local support. This chapter deals with important computational aspects related to sparsity: storage, assembling, and reorderi...

This chapter presents a step-by-step derivation of weak formulations. We start by considering a few simple PDEs posed over a bounded domain, and we reformulate these problems in weak form using the important notion of test functions. We show by examples that there are many ways to write weak formulations. We consider second-order and first-order PD...

This chapter addresses fundamental properties of scalar-valued second-order elliptic PDEs endowed with a coercivity property. The prototypical example is the Laplacian with homogeneous Dirichlet conditions. More generally, we consider PDEs including lower-order terms, such as the diffusion-advection-reaction equation, where the lower-order terms ar...

In this chapter, we study the Galerkin approximation of the model problem considered in the previous two chapters. We focus on the well-posedness of the approximate problem, and we derive a bound on the approximation error in a simple setting. This bound is known in the literature as Céa’s lemma. We also characterize the well-posedness of the discr...

The present chapter contains a brief introduction to the spectral theory of compact operators together with illustrative examples. Eigenvalue problems occur when analyzing the response of devices, buildings, or vehicles to vibrations, or when performing the linear stability analysis of dynamical systems.

In this chapter, we study some further questions regarding the approximation of second-order elliptic PDEs by H1-conforming finite elements: (i) How can non-homogeneous Dirichlet conditions be taken into account in the error analysis, and how can they be implemented in practice; (ii) Can the discrete problem reproduce the maximum principle; (iii) H...

The objective of this chapter is to give a brief overview of the analysis of the Helmholtz problem and its approximation using H1-conforming finite elements. The Helmholtz problem arises when modeling electromagnetic or acoustic scattering problems in the frequency domain. One specificity of this elliptic problem is that one cannot apply the Lax-Mi...

In this chapter, we continue the study of stable finite element pairs that are suitable to approximate the Stokes equations. In doing so, we introduce another technique to prove the inf-sup condition that is based on a notion of macroelement. We more specifically focus on the case where the discrete pressure space is a broken finite element space.

The objective of this chapter is to introduce some model problems derived from Maxwell’s equations that all fit the Lax-Milgram formalism. The approximation is performed using edge (Nédélec) finite elements. The analysis relies on a coercivity argument in H(curl) that exploits the presence of a uniformly positive zero-order term in the formulation....

This chapter reviews various stable finite element pairs that are suitable to approximate the Stokes equations. We first review two standard techniques to prove the inf-sup condition, one based on the Fortin operator and one hinging on a weak control of the pressure gradient. Then we show how these techniques can be applied to finite element pairs...

In this chapter, we consider Darcy’s equations as the simplest example of elliptic PDEs written in mixed form. We derive well-posed weak formulations with various boundary conditions. Then we study mixed finite element approximations using H(div)-conforming spaces for the dual variable.

The objective of this chapter is to study the approximation of eigenvalue problems associated with symmetric coercive differential operators using H1-conforming finite elements. The goal is to derive error estimates on the eigenvalues and the eigenfunctions.

The goal of this chapter and the next one is to investigate the approximation of a diffusion model problem with contrasted diffusivity and revisit the error analysis of the various nonconforming approximation methods presented in the previous chapters. The essential difficulty is that the elliptic regularity theory tells us that the Sobolev smoothn...

In this chapter, we first show that the discrete problem generated by the Galerkin approximation can be reformulated as a linear system once bases for the discrete trial space and the discrete test space are chosen. Then, we investigate important properties of the system matrix, which is called stiffness matrix. We also introduce the mass matrix, w...

This chapter is concerned with the approximation of the model problem analyzed in the previous chapter. We focus on the Galerkin approximation in the conforming setting. We establish necessary and sufficient conditions for well-posedness, and we derive error bounds in terms of the best-approximation error. Then we consider the algebraic viewpoint,...

This chapter departs from the ideal setting analyzed in the previous chapter. The approximation is no longer conforming, and the consequences of various so-called variational crimes are studied. The main results of this chapter are upper bounds on the approximation error in terms of the best-approximation error of the exact solution by members of t...

The objective of the present chapter is to study the nonconforming approximation of the Poisson model problem by Crouzeix–Raviart finite elements. In doing so, we illustrate the abstract error analysis from Chapter 27.

The Stokes equations constitute the basic linear model for incompressible fluid mechanics. We first derive a weak formulation of the Stokes equations and establish its well-posedness. The approximation is then realized by means of mixed finite elements, that is, we consider a pair of finite elements, where the first component of the pair is used to...

In this chapter, we continue our investigation of the finite element approximation of eigenvalue problems, but this time we do not assume symmetry and we explore techniques that can handle nonconforming approximation settings.

The objective of this chapter is to describe techniques for the solution of hyperbolic systems that are at least (informally) second-order accurate in space and invariant domain preserving. As seen in the previous chapter, one can make the method more accurate in space by decreasing the first-order graph viscosity. Another technique, which is very...

In this chapter, we study a time-stepping technique for the time-dependent Stokes equations based on an artificial compressibility perturbation of the mass conservation equation. This technique presents some advantages with respect to the projection methods. It avoids solving a Poisson equation at each time step, and it can be extended to high orde...

Employing inf-sup stable mixed finite elements to solve Stokes-like problems may seem to be a cumbersome constraint. The goal of this chapter is to show that it is possible to work with pairs of finite elements that do not satisfy the inf-sup condition provided the Galerkin formulation is slightly modified. This is done by extending to the Stokes p...

In this chapter, we introduce the prototypical model problem for first-order PDEs: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs. This system enjoys symmetry and positivity properties and, in the literature, it is often referred to as Friedrichs’ system. Friedrichs’ formalism is very powerful and encompasses several mod...

In this chapter, we want to solve a model problem where the PDE comprises a first-order differential operator modeling advection processes and a second-order term modeling diffusion processes. The difficulty in approximating an advection-diffusion equation can be quantified by the Péclet number which is equal to the meshsize times the advection vel...

The goal of this chapter is to derive a weak formulation of a model parabolic problem and to establish its well-posedness. The prototypical example is the heat equation. To this purpose, we use the Bochner integration theory presented in the previous chapter.

In this chapter, we continue the unified analysis of fluctuation-based stabilization techniques for Friedrichs’ systems. We now focus on two closely related stabilization techniques known in the literature as local projection stabilization (LPS) and subgrid viscosity (SGV). The key idea is to introduce a two-scale decomposition of the discrete \(H^...

In this chapter, instead of using stabilized \(H^1\)-conforming finite elements, we consider the discontinuous Galerkin (dG) method. The stability and convergence properties of the method rely on choosing a numerical flux across the mesh interfaces. Choosing the centered flux yields suboptimal convergence rates for smooth solutions. The stability p...

This chapter gives a brief overview of some splitting techniques to approximate the time-dependent Stokes problem in time. The common feature of the algorithms is that each time step leads to subproblems where the velocity and the pressure are uncoupled. The linear algebra resulting from the space approximation is therefore simplified, making these...

We continue in this chapter the study of stabilization techniques to approximate the Stokes problem with finite element pairs that do not satisfy the inf-sup condition. We now focus our attention on the continuous interior penalty and the discontinuous Galerkin methods.

In this chapter, we discuss two time-stepping techniques that deliver second-order accuracy in time and, like the implicit Euler method, are unconditionally stable. One technique is based on the second-order backward differentiation formula (BDF2), and the other, called Crank–Nicolson, is based on the midpoint quadrature rule. Since the BDF2 method...

The goal of this chapter is to introduce a mathematical setting to formulate parabolic problems in some weak form. The viewpoint we are going to take is to consider functions defined on a bounded time interval with values in some Banach (or Hilbert) space composed of functions defined on the space domain. The key notions we develop in this chapter...

The present chapter deals with the approximation of the time-dependent Stokes equations. We use stable mixed finite elements for the space discretization in a conforming setting. The time discretization can be done with any of the techniques considered for the heat equation. For brevity, we focus on the implicit Euler scheme and on higher-order imp...

This chapter focuses on the approximation of nonlinear hyperbolic systems using finite elements. We describe a somewhat loose adaptation to finite elements of a scheme introduced by Lax. The method, introduced by Guermond, Nazarov, and Popov, can be informally shown to be first-order accurate in time and space and to preserve every invariant set of...

In this chapter, we consider the same space semi-discrete problem as in the previous chapter, but we now discretize it in time using an explicit scheme. We first discuss generic properties of explicit Runge–Kutta schemes (ERK). Then we analyze the explicit Euler scheme, second-order two-stage ERK schemes, and third-order three-stage ERK schemes. Th...

In this chapter, we continue the study started in the previous chapter on higher-order time approximation schemes using a space-time functional framework. The trial functions are now continuous in time and piecewise polynomials with a polynomial degree that is one order higher than that of the test functions. The resulting technique is called conti...

The goal is now to discretize in time the space semi-discrete parabolic problem considered in the previous chapter. Since this problem is a system of coupled (linear) ODEs, its time discretization can be done by using one of the numerous time-stepping techniques available from the literature. In this chapter, we focus on the implicit (or backward)...

In this chapter, we still use \(H^1\)-conforming finite elements and a boundary penalty technique, but we consider a different stabilization technique. One motivation is that the residual-based stabilization from the previous chapter is delicate to use when approximating time-dependent PDEs since the time derivative is part of the residual. The tec...

This chapter gives a brief description of the theory of scalar conservation equations. We introduce the notions of weak and entropy solutions and state existence and uniqueness results. Even if the initial data is smooth, the solution of a generic scalar conservation equation may lose smoothness in finite time, and weak solutions are in general non...

The objective of this chapter is to introduce the concept of hyperbolic systems and to generalize the notions introduced in the previous chapter to this class of equations. The novelty here is that the notion of maximum principle is no longer valid and is replaced by the concept of invariant sets.

This chapter is concerned with the approximation of Friedrichs’ systems using \(H^1\)-conforming finite elements. The main issue one faces in this context is to achieve stability. The stabilization techniques presented in this chapter are inspired by the least-squares (LS), or minimal residual, technique from linear algebra. The LS approximation gi...

In this chapter, we consider time-dependent Friedrichs’ systems. The prototypical example is the linear transport equation. We first derive a functional setting for the model problem and establish its well-posedness. For simplicity, we assume that the differential operator in space is time-independent. Then we construct a space semi-discretization...

We are concerned in this chapter with the semi-discretization in space of the model parabolic problem introduced in the previous chapter. We use conforming finite elements for the space approximation. Error estimates are derived by invoking coercivity-like arguments. Semi-discretization in space leads to a (large) system of coupled ordinary differe...

This book is the third volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedica...

This chapter focuses on the weak formulation of the time-dependent Stokes equations. We consider two possible weak formulations. The first one enforces the divergence-free constraint on the velocity field without introducing the pressure. This formulation can be handled by using the same analysis tools as for parabolic problems. The second weak for...

The objective of this chapter is to describe techniques that preserve the invariant domain property of the algorithm introduced in the previous chapter and increase its accuracy in time and space. The argumentation for the time approximation is done for general hyperbolic systems, but the argumentation for the space approximation is done for scalar...

In the previous two chapters, we have used finite differences to approximate the time derivative in the space semi-discrete parabolic problem. We now adopt a different viewpoint directly relying on a space-time weak formulation. The time approximation is realized by using piecewise polynomial functions over the time mesh. The test functions are dis...

In this chapter, we continue the study of the time-dependent Friedrichs’ systems. In the previous chapter, we have established the well-posedness of the continuous model problem and we have discretized the problem in space using \(H^1\)-conforming finite elements, a boundary penalty technique, and fluctuation-based stabilization. In this chapter, w...

In this chapter, we revisit the well-posedness of the model parabolic problem, i.e., we give another proof of Lions’ theorem using the framework of the BNB theorem. In other words, we establish the well-posedness by proving an inf-sup condition. Then we exploit the inf-sup condition to revisit the stability and the error analysis for various approx...

The benefit of transformer oil seeded with magnetic nanoparticles (forming a transformer oil-based ferrofluid) is numerically studied in an axisymmetric model of 40 kVA transformer. The maximum temperature in the system is decreased by 2.2°C when using ferrofluid instead of transformer oil. This effect is mainly due to the magnetic body force, whic...

We present a fully discrete approximation technique for the compressible Navier–Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, i.e. τ≲O(h)∕V where V is some reference velocity scale and h the...

The objective of this note is to propose a relaxation technique that accounts for the topography effects in the dispersive Serre equations (also known as Serre--Green--Naghdi or fully non-linear Boussinesq equations, etc.) introduced in [https://doi.org/10.1017/S0022112087000594]. This is done by revisiting the relaxation technique introduced in [h...

This book is the first volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedica...

In this chapter and the next one, we study the interpolation operators associated with the finite elements introduced in Chapters 14 and 15. We consider a shape-regular sequence of affine simplicial meshes with a generation-compatible orientation. In the present chapter, we show how the degrees of freedom attached to the faces and the edges can be...

This chapter introduces key notions for finite elements, such as degrees of freedom, shape functions, and interpolation operator. These notions are illustrated on Lagrange finite elements and modal finite elements, for which the degrees of freedom are values at specific nodes and moments against specific test functions, respectively.

This chapter deals with finite elements defined on a simplex (triangle in 2D, tetrahedron in 3D). The degrees of freedom are either nodal values at some points on the simplex or integrals over the faces or the edges of the simplex, and the associated functional space is composed of multivariate polynomials of prescribed total degree. We focus our a...

The quasi-interpolation operators introduced in Chapter 22 are \(L^1\)-stable, are projections and have optimal (local) approximation properties. However, they do not commute with the usual differential operators (gradient, curl, and divergence), which makes them difficult to use to approximate simultaneously a vector-valued function and its curl o...

This chapter presents important examples of finite elements, first in dimension one, then in multiple dimensions using tensor-product techniques. Important computational issues related to the manipulation of high-order polynomial bases are addressed. We also show how to approximate integrals over intervals using the roots of the Legendre and Jacobi...

The goal of this chapter is to construct vector-valued finite elements to approximate fields with integrable curl. The finite elements introduced in this chapter can be used, e.g., to approximate (simplified forms of) Maxwell’s equations which constitute a fundamental model in electromagnetism. The focus here is on defining a reference element and...

Inverse inequalities rely on the fact that all the norms are equivalent in finite-dimensional normed vector spaces. The term ‘inverse’ refers to the fact that high-order Sobolev (semi)norms are bounded by lower-order (semi)norms, but the constants involved in these estimates either tend to zero or to infinity as the meshsize goes to zero. Our purpo...

The goal of this chapter and the following one is to construct the connectivity array introduced in the previous chapter. In this chapter, we see how to enforce the zero-jump condition across every mesh interface by means of the degrees of freedom on the two mesh cells sharing this interface. In particular, we identify two key structural assumption...

We analyze the local finite element interpolation error for smooth functions. We restrict the material to affine meshes and to relatively simple functional transformations. We introduce the notion of shape-regular families of affine meshes, we study the transformation of Sobolev norms, and we present important approximation results collectively kno...

We show how to generate a finite element in each mesh cell from a reference finite element. To this purpose, we need one new concept in addition to the geometric mapping: a functional transformation that maps functions defined on the current mesh cell to functions defined on the reference cell. Key examples of such transformations are the Piola tra...

We study how to build a mesh of a bounded domain, i.e., a finite collection of cells forming a partition of the domain. Building a mesh is the first important task to realize when one wants to approximate some PDEs posed in the domain. The viewpoint we adopt in this book is that each mesh cell is the image of a reference cell by some smooth diffeom...

In this chapter, we extend the results of Chapter 11 to nonaffine meshes. For simplicity, the functional transformation is the pullback by the geometric mapping, but this mapping is now nonaffine. The first difficulty consists of comparing Sobolev norms. This is not a trivial task since the chain rule involves higher-order derivatives of the geomet...

In this chapter, we continue our investigation of the interpolation operators associated with the finite elements introduced in Chapters 14 and 15. We consider a shape-regular sequence of affine simplicial meshes with a generation-compatible orientation. The key idea here is to extend the degrees of freedom on the faces and the edges by requiring s...

One of the objectives of this chapter is to estimate the decay rate of the best-approximation errors of functions in Sobolev spaces by members of conforming finite element spaces. The interpolation operators constructed so far do not give a satisfactory answer to the above question when the functions have a low smoothness index. In this chapter, we...

The objective of the four chapters composing Part I of this book is to recall (or gently introduce) some elements of functional analysis that will be used throughout the book: Lebesgue integration, weak derivatives, and Sobolev spaces. We focus in this chapter on Lebesgue integration and Lebesgue spaces. Most of the results are stated without proof...

## Citations

... A fundamental mathematical property of any reasonable numerical scheme for nonlinear systems of hyperbolic conservation laws that was not investigated in this paper at all was the invariant domain preserving property (IDP), such as the positivity of density and entropy. Future work will consider provably IDP extensions of our methods, making use of the mathematical techniques presented in the seminal work of Guermond and Popov et al. concerning provably invariant domain preserving schemes [31,[57][58][59] and in the work of Kuzmin et al. [7,[65][66][67] concerning bound-preserving algebraic flux limiters and slope limiters for high order continuous and discontinuous Galerkin finite element schemes. ...

... • In [9], well-balanced schemes for sediment transport are introduced. • Second-order ALE schemes are described in [10]. • Two-phase oil-water movement is studied in [13]. ...

... We use entropy-dissipative semidiscretizations to ensure robustness [29,31,83,86] but do not investigate specific implementation techniques discussed elsewhere [33,61,79]. Instead, we focus on step size control of time integration methods applied to ODEs resulting from spatial semidiscretizations of compressible flow problems. ...

... The idea is to construct a method that is at least second-order accurate in space, third order accurate in time, well balanced and invariant-domain preserving (i.e., robust with respect to dry states). The starting point of this present paper is the hyperbolic relaxation technique introduced in [12] and further expanded in [23] for solving the dispersive Serre-Green-Naghdi equations with topography effects. The approach reformulates the Serre-Green-Naghdi equations as a first-order hyperbolic system which allows for explicit time stepping. ...

... In the spirit of Buffa & Perugia (2006), Bonito et al. (2016) proposed an interior-penalty method with C 0 finite elements for the Maxwell equations with minimal smoothness requirements. Recently, Ern & Guermond (2021) analyzed a nonconforming approximation of elliptic partial differential equations (PDEs) with minimal regularity by introducing a generalized normal derivative of the exact solution at the mesh faces. They also showed that this idea can be extended to solve the time-harmonic Maxwell equations with low regularity solutions by introducing a more general concept for the tangential trace. ...

... Discontinuous Galerkin finite element method (DGFEM) was first introduced by Reed and Hill [16] in 1973 and has been developed greatly (see, e.g., [17][18][19][20][21][22][23][24]). DGFEM for eigenvalue problems has also been discussed in many papers (see [25][26][27][28][29][30][31][32][33][34]). ...

... Remark 1 (Tensor spaces). Bochner spaces can be seen as closure of algebraic tensor product spaces [EG21b,Rem. 64.24], i.e., for V ∈ {H 1 0 (Ω), L 2 (Ω); ...

... Mass conservation (Mills, 1999): ...

... The peak magnetic field intensity in power transformers can reach few hundreds of kA/m depending on the geometry of the coil assembly [20], [21]. The operational magnetic flux density in the transformer core is generally close to 1.5T and the leakage flux density is around 0.15T [22], [23]. ...

... Mass transport effects have, to a certain extend, been studied by numerical simulation -but almost exclusively within the cathode. Investigated phenomena include pure diffusion [3,50], thermal convection [8,51], solutal convection [52][53][54][55] and electro-vortex flow [11,54,[56][57][58]. With the exception of one model [50,56], the influence of the flow on the cell voltage has always been strongly simplified. ...