We construct a non-polynomial local discontinuous Galerkin (LDG) scheme for the prescribed mean curvature equation to approximate boundary gradient blow-up solutions, and obtain error estimates.

This work introduces and analyzes B-spline approximation spaces defined on general geo- metric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, em- ploying either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive ap- proximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution—particularly concerning isotropic (non-mixed) derivatives—are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor- product domains.

We study an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes using the CutFEM method. Our main contribution is the reconstruction of an element-wise conservative flux from the CutFEM solution and its use in a posteriori error estimation. We introduce a hybrid mixed formulation with locally computable Lagrange multipliers and reconstruct an equilibrated flux in the immersed Raviart-Thomas space. Based on this, we propose a new a posteriori error estimator that includes both volume and interface terms. We state its robust reliability and local efficiency, which are proved in Part II of this work. The approach is validated through numerical experiments.

For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modification of a positivity preserving fully discrete scheme using a local extremum diminishing flux limiter. We discretize space using piecewise linear finite elements on an quasiuniform triangulation of acute type and time by the backward Euler method. We assume that initial data are sufficiently small in order not to have a blow-up of the solution. Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show existence of the fully discrete approximation and derive error bounds in $L^2$ for the cell density and $H^1$ for the chemical concentration. We also present numerical experiments to illustrate the theoretical results.

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.