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 methods have gained increasing attention for solving large-scale inverse problems due to their computational efficiency. However, their theoretical justification in the context of ill-posed problems remains underdeveloped, particularly regarding convergence rate analysis, where existing results typically yield only suboptimal rates. In this paper, we address this gap by establishing order-optimal convergence rates for a stochastic gradient method applied to linear ill-posed problems in Hilbert spaces. Under Hölder-type source conditions with smoothness parameter ν∈ (0, 1/2] , we derive convergence rates both in expectation and almost surely, accommodating a broad class of step-size sequences, including constant and polynomially decaying ones. Our analysis is based on a delicate Lyapunov-type argument and an application of the Robbins–Siegmund theorem. As a byproduct, we also establish new convergence results that do not rely on any source conditions.
Hierarchical Model (HiMod) reduction is an effective Reduced Order Modelling technique for problems defined on elongated, pipe-like domains. It is particularly suitable when a dominant dynamics is aligned with the longitudinal direction, while transverse effects are locally significant but spatially limited. When applied to two-field problems such as the Stokes equations, a main challenge is to ensure the stability of the reduced formulation, particularly the inf-sup condition for pressure discretization. In this work, we provide a rigorous analysis showing that the inf-sup condition holds whenever the number of velocity modes is at least equal to the number of pressure modes, thereby extending previous heuristic approaches. The proof exploits the separation of variables in HiMod and is valid for pipe-like domains under some geometric assumptions. Numerical assessment confirms the theoretical findings, providing a solid foundation for stable and efficient HiMod reduction in incompressible flow problems.
We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov–Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in L^∞ -type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the L^∞ (L^2) norm with fully explicit constants; to this aim, we design a reconstruction operator into 𝒞^1 piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings.
The classical arguments employed when obtaining error estimates of Finite Element (FE) discretisations of elliptic problems lead to more restrictive assumptions on the regularity of the exact solution when applied to non-conforming methods. The so-called minimal regularity estimates available in the literature relax some of these assumptions, but are not truly of minimal regularity , since a data oscillation term appears in the error estimate. Employing an approach based on a smoothing operator, we derive for the first time error estimates for Discontinuous Galerkin (DG) type discretisations of non-linear problems with $$(p,\delta )$$ ( p , δ ) -structure that only assume the natural $$W^{1,p}$$ W 1 , p -regularity of the exact solution, and which do not contain any oscillation terms.
