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1. Nash Equilibrium in Discontinuous Games

   https://doi.org/10.1146/annurev-economics-082019-111720  

   Reny, Philip J. (August 2020, Annual Review of Economics)

   We review the discontinuous games literature, with a sharp focus on conditions that ensure the existence of pure and mixed strategy Nash equilibria in strategic form games and of Bayes-Nash equilibria in Bayesian games. 

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2. Half-closed discontinuous Galerkin discretisations

   https://doi.org/10.1016/j.jcp.2025.113731  

   Pan, Y; Persson, P-O (March 2025, Journal of Computational Physics)
3. The discontinuous Petrov–Galerkin method

   https://doi.org/10.1017/S0962492924000102  

   Demkowicz, Leszek; Gopalakrishnan, Jay (July 2025, Acta Numerica)

   The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research on theory and applications of the DPG methodology. The scope of the presented theory is restricted to linear problems on Hilbert spaces, but pointers to extensions are provided. Multiple viewpoints to the basic theory are provided. They show that the DPG method is equivalent to a method which minimizes a residual in a dual norm, as well as to a mixed method where one solution component is an approximate error representation function. Being a residual minimization method, the DPG method yields Hermitian positive definite stiffness matrix systems even for non-self-adjoint boundary value problems. Having a built-in error representation, the method has the out-of-the-box feature that it can immediately be used in automatic adaptive algorithms. Contrary to standard Galerkin methods, which are uninformed about test and trial norms, the DPG method must be equipped with a concrete test norm which enters the computations. Of particular interest are variational formulations in which one can tailor the norm to obtain robust stability. Key techniques to rigorously prove convergence of DPG schemes, including construction of Fortin operators, which in the DPG case can be done element by element, are discussed in detail. Pointers to open frontiers are presented. 

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4. Dune dynamics drive discontinuous barrier retreat

   https://doi.org/doi.org/10.1029/2021GL092958  

   Reeves, I. R. (January 2021, Geophysical research letters)

   null (Ed.)
5. Dune Dynamics Drive Discontinuous Barrier Retreat

   https://doi.org/10.1029/2021GL092958  

   Reeves, I. R. B.; Moore, L. J.; Murray, A. B.; Anarde, K. A.; Goldstein, E. B. (July 2021, Geophysical Research Letters)

   Abstract Barrier islands and spits tend to migrate landward in response to sea‐level rise through the storm‐driven process of overwash, but overwash flux depends on the height of the frontal dunes. Here, we explore this fundamental linkage between dune dynamics and barrier migration using the new model Barrier3D. Our experiments demonstrate that discontinuous barrier retreat is a prevalent behavior that can arise directly from the bistability of foredune height, occurring most likely when the storm return period and characteristic time scale of dune growth are of similar magnitudes. Under conditions of greater storm intensity, discontinuous retreat becomes the dominant behavior of barriers that were previously stable. Alternatively, higher rates of sea‐level rise decrease the overall likelihood of discontinuous retreat in favor of continuous transgression. We find that internal dune dynamics, while previously neglected in exploratory barrier modeling, are an essential component of barrier evolution on time scales relevant to coastal management. 

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