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

... 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. ...