Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.02946

A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems

Sheraz Ahmed Khan·Ramon Codina·Hauke Gravenkamp
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional $Q(u)$ of the unknown $u$. If the SGS $u'$ is estimated, the error in the quantity of interest can be approximated by $Q(u')$. Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sheraz Ahmed Khan, Ramon Codina, Hauke Gravenkamp. 2025-05-05. A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems. https://arxiv.org/abs/2505.02946

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Discrete normalized gradient flow for two-component Bose-Einstein condensates: Energy dissipation, global convergence and sharp local convergence behavior

The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. We apply GFSI to the two-component scenario with Josephson junction and rotating term, which is one of the most important and topical models in multi-component Bose-Einstein condensates (MBECs), and rigorously establish the following fundamental results for the first time. By introducing a Lagrange multiplier to reformulate GFSI into an equivalent form, we prove its energy dissipation property and global convergence to stationary states. More significantly, we uncover an intrinsic connection between this classical numerical PDE discretization rooted in imaginary-time evolution and Riemannian optimization, a state-of-the-art mathematical framework for manifold-constrained optimization. This connection enables us to fully characterize the local convergence behavior of GFSI within the Riemannian optimization framework. Together with the aforementioned global convergence result, this yields a complete global--local convergence theory for GFSI. Finally, numerical experiments comprehensively validate the theoretically predicted energy dissipation and convergence properties.

math.NA↗
arXiv2026

Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the $L^p$-Wasserstein distance ($p\geqslant 1$). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time $1/2$-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a $1/2$-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the $L^1$-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

math.NA↗
arXiv2026

Barotropic-Baroclinic Splitting for Multilayer Shallow Water Models with Exchanges

This work presents the numerical analysis of a barotropic-baroclinic splitting in a nonlinear multilayer framework with exchanges between the layers in terrain-following coordinates. The splitting is formulated as an exact operator splitting. The barotropic step handles free surface evolution and depth-averaged velocity via a well-balanced one-layer model, while the baroclinic step manages vertical exchanges between layers and adjusts velocities to their mean values. We show that the barotropic-baroclinic splitting preserves total energy conservation and meets both a discrete maximum principle and a discrete entropy inequality. Several numerical experiments are presented showing the gain in computational cost, particularly in low Froude simulations, with no loss of accuracy. The benefits of using a well-balancing strategy in the barotropic step to preserve the geostrophic equilibrium are inherited in the overall scheme.

math.NA↗