The reduction factor ϱ in a Galerkin scheme with conforming P_1 finite element spaces on simplicial meshes is the maximal quotient of the Galerkin errors in the L^2 norm divided by the energy norm (the H^1 semi-norm) for the Poisson model problem for instance. An old and a new scheme allow for the computation of upper bounds of ϱ on a given mesh. The schemes are compared theoretically and numerically before three applications for the guaranteed lower eigenvalue bounds (GLB) illustrate their impact. The three eigenvalue problems are named after Laplace, Schrödinger, and Steklov. The resulting GLB compete with those from recent post-processing of nonconforming Crouzeix–Raviart eigenvalue approximations in the classical and in extended versions.
Stochastic gradient type methods, including stochastic mirror descent, have become fundamental tools for solving large-scale ill-posed inverse problems. However, when combined with general strongly convex penalties for feature-enhancing reconstruction, these methods suffer from slow convergence, oscillatory behavior, and pronounced semi-convergence effects caused by data noise and ill-posedness. To address these issues, we propose an Adaptive Stochastic Nesterov (ASN) method for solving inverse problems with multiple operators under general strongly convex regularization. The method integrates a general convex functional into a stochastic framework and introduces an adaptive Nesterov-type acceleration mechanism in which both the step size and momentum parameters are computed automatically through explicit iterative formulas, avoiding manual tuning. A rigorous a posteriori stopping rule is developed, ensuring finite termination and stability of the reconstruction. Theoretical analysis proves the well-definedness of the iteration and establishes monotonic energy decay and stability properties. Numerical experiments demonstrate that the proposed method significantly accelerates convergence while better preserving structural features such as sparsity and discontinuities.
We construct a numerical solution, based on a Fourier spectral method to the spatially homogeneous Landau equation with Coulomb potential in a bounded domain 𝒟_L . By deriving an explicit error estimate in terms of the length of the domain L and the number of Fourier modes 2N, we demonstrate that for any prescribed error tolerance and fixed time interval [0, T], there exist choices of 𝒟_L and N satisfying explicit conditions such that the error between the numerical and exact solutions is below the tolerance. Specifically, the estimate shows that sufficiently large L and N (depending on initial data parameters and T) can reduce the error to any desired level. Numerical simulations based on this construction are also presented.
We present a convergent and constraint-preserving fully-discrete finite element discretization of a mathematical model for the dynamics of a liquid crystal subjected to an electric field. This model can be derived from the Oseen-Frank director field theory, assuming that the dynamics of the electric field are governed by the electrostatics equations with a suitable constitutive relation for the electric displacement field that describes the coupling with the liquid crystal director field. The resulting system of partial differential equations consists of an elliptic equation that is coupled to the wave map equations through a quadratic source term. We show that the discretization preserves the unit length constraint of the director field at every node of the mesh, is energy-stable and convergent. In numerical experiments, we show that the method is stable even when singularities develop. Moreover, predictions about the alignment of the director field with the electric field are confirmed.
We present a wavenumber-explicit analysis of the FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in [1] for conforming discretizations and in [2] for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that kh/p be sufficiently small and that log (k) / p be bounded imply quasi-optimality of both conforming and DG-method, where k is the wavenumber, h the mesh size, and p the approximation order. The analysis relies on a k-explicit regularity theory for a three-field coupling formulation.
