Abstract. We study a class of semi-discrete variational problems that arise in economic matching and game theory, where agents with continuous attributes are matched to a finite set of outcomes with a one dimensional structure. Such problems appear in applications including Cournot-Nash equilibria, and hedonic pricing, and can be formulated as problems involving optimal transport between spaces of unequal dimensions. In our discrete strategy space setting, we establish analogues of results developed for a continuum of strategies in [25], ensuring solutions have a particularly simple structure under certain conditions. This has important numerical consequences, as it is natural to discretize when numerically computing solutions. We adapt standard semi-discrete optimal transport techniques to the variational setting in which the target measure is unknown. By leveraging discrete nestedness when it holds, our sequential algorithms improve robustness and achieve computational gains, together with rigorous convergence guarantees, as demonstrated through numerical experiments.

We address the stabilization of linear, time-varying parabolic PDEs using finite-dimen\-sional receding horizon controls (RHCs) derived from reduced-order models (ROMs). We first prove exponential stability and suboptimality of the continuous-time full-order model (FOM) RHC scheme in Hilbert spaces. A Galerkin model reduction is then introduced, along with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This results in a ROM-based RHC algorithm that adaptively constructs reduced-order controls, ensuring exponential stability of the FOM closed-loop state and providing computable performance bounds with respect to the infinite-horizon FOM control problem. Numerical experiments with a non-smooth cost functional involving the squared $\ell^1$-norm confirm the method’s effectiveness, even for exponentially unstable systems.

In this work, we derive a hyperbolic system of dispersive equations for the numerical simulation of coastal waves with improved dispersive properties and admitting an exact energy conservation equation. This system is derived with the assumption of a moderate non-linearity and of a correction coefficient close to 1. This system contains the same non-linear terms as the Serre-Green-Naghdi equations, which are obtained in the limit where the Mach number tends to zero. The assumptions are only used to neglect non-linear terms related to the improvement of dispersive properties. The bathymetry can be included with a mild-slope hypothesis. On this basis, we propose an energy-stable numerical scheme relying on a splitting between the hyperbolic and dispersive parts of the model. The stability of the method is achieved through the discrete dissipation of the energy balance specific to each step. We also establish the existence of soliton solutions for this model. Numerical simulations are proposed to highlight the dispersive properties of the model, as well as the dissipative character of the scheme

We study finite-difference approximations of the Poisson–Boltzmann (PB) electrostatic energy functional of ionic concentrations and electric displacements constrained by Gauss’ law and the ionic mass conservation, and a class of local algorithms for minimizing the finite-difference discretized such energy functional. We prove that the discrete Boltzmann distributions characterize the finite-difference minimizer and obtain the uniform bounds and optimal error estimates in maximum norm for such a minimizer. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy by updating the concentrations and displacement one grid box at a time, keeping Gauss’ law and the mass conservation satisfied. A new local algorithm with a shift is constructed for minimizing the Poisson electrostatic energy (the part of the PB energy without ionic concentrations) with a variable dielectric coefficient. We prove the convergence of these local algorithms and present numerical tests to demonstrate the results of our analysis.

In this work, a balancing domain decomposition by constraints (BDDC) algorithm is applied to the nonsymmetric positive definite linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of an elliptic distributed optimal control problem. Convergence analysis for the BDDC preconditioned generalized minimal residual (GMRES) solver demonstrates that, when the subdomain size is small enough, the algorithm is robust with respect to the regularization parameter, and the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. Numerical experiments are performed to confirm the theoretical results.

In this paper, we propose and analyze a strongly mass-conservative numerical scheme for the coupled Navier--Stokes and Darcy--Forchheimer system in both two and three spatial dimensions. The two subproblems are coupled through physically relevant interface conditions, including mass conservation, balance of normal forces, and the Beavers--Joseph--Saffman condition. We employ a staggered discontinuous Galerkin method for the Navier-Stokes equations and use standard mixed finite elements for the Darcy-Forchheimer problem. The proposed formulation incorporates the interface conditions directly, without introducing Lagrange multipliers on the interface or artificial numerical fluxes on the mesh skeleton. As a consequence, although discontinuous Galerkin elements are used in the free-flow region, the resulting discrete velocity field is globally $\bm{H}(\mathrm{div})$-conforming across the entire domain. In particular, the incompressibility constraint is satisfied exactly in the free-flow region, thereby yielding strong mass conservation over the entire computational domain. Under a suitable small-data assumption, we establish the well-posedness of the resulting nonlinear discrete system. Owing to the exact preservation of mass conservation, the proposed scheme exhibits a pressure-robust behavior, in the sense that the velocity approximation is insensitive to pressure effects. Numerical experiments are presented to illustrate the stability and robustness of the method, including its performance in regimes involving small viscosity, large pressure, and limited solution regularity.

In this work, we propose an improved discretization, in terms of stability and accuracy, for the incompressible two-phase Darcy flows in a heterogeneous porous medium with discontinuous capillary forces. For this purpose, the total velocity formulation of the model is used. The coupled system is composed of a degenerate parabolic equation for the non-wetting phase and a pressure equation for the total velocity. We combine a positive Vertex Approximation Gradient (VAG) type scheme for the gradient fluxes with a hybrid upwinding of the mobilities. This approach entails a maximum principle on the saturations, which remain in their physical ranges. Energy estimates are obtained by selecting key approximations of the fluxes. These stability results allow to prove the existence of discrete solutions. Numerical experiments on complex test-cases show the robustness of the new approach in terms of the accuracy as well as the nonlinear convergence. Comparison to the usual phase potential upwinding approach and to a previous hybrid upwinding scheme are also provided.

We construct and analyze a projection-free linearly implicit method for the approximation of flows of harmonic maps into spheres. The proposed method is unconditionally energy stable and, under a sharp discrete regularity condition, achieves second-order accuracy with respect to the constraint violation. Furthermore, the method accommodates variable step sizes to speed up the convergence to stationary points and to improve the accuracy of the numerical solutions near singularities, without affecting the unconditional energy stability and the constraint violation property. We illustrate the accuracy in approximating the unit-length constraint and the performance of the method through a series of numerical experiments, and compare it with the linearly implicit Euler and two-step BDF methods.

We study the properties of a semi-implicit Euler scheme that is widely used in time discretization of Keller-Segel equations both in the parabolic-elliptic form and the parabolic-parabolic form. \bl{ We assume here the smallness of the mass of the initial data in order to ensure that the solutions of the original Keller-Segel equations do not blow up in finite time}. We prove that this linear, decoupled, first-order scheme preserves the important properties of Keller-Segel equations at the semi-discrete level, including the mass conservation and positivity preserving of the cell density, and the energy dissipation. We also establish optimal error estimates in $L^p$-norm $(1<p<\infty)$.

We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities, perturbed by spatially smooth multiplicative Gaussian noise. The proposed scheme is applicable to a range of physically relevant nonlinear models, including the stochastic Landau--Lifshitz--Baryakhtar (sLLBar) equation, the stochastic convective Cahn--Hilliard equation with mass source, and the stochastic regularised Landau--Lifshitz--Bloch (sLLB) equation, among others. To overcome the difficulties posed by the interplay between the nonlinearities and the stochastic forcing, we adopt a `truncate-then-discretise' strategy: the nonlinear term is first truncated before discretising the resulting modified problem. We show that the strong solution to the truncated system converges in probability to that of the original problem. A fully discrete numerical scheme is then proposed for the truncated problem. Assuming initial data in $\mathbb{H}^2$, we utilise parabolic smoothing estimates and the temporal H\"older continuity of the solution to establish both convergence in probability and strong convergence (with quantitative rates) for the two fields used in the mixed formulation. Numerical simulations are provided to support the theoretical results.