Residual-based stabilized reduced-order models of the transient convection-diffusion-reaction equation obtained through discrete and continuous projection
Eric Parish, Masayuki Yano, Irina Tezaur, Traian Iliescu
Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for advection-dominated systems when the ROM is under-resolved. While these two approaches can be equivalent in certain settings, differing techniques have emerged in both contexts. This work outlines these two approaches within the setting of finite element method (FEM) discretizations (in which case a duality exists between the continuous and discrete levels) of the convection-diffusion-reaction equation, and compares residual-based stabilization techniques that have been developed in both contexts. In the spatially continuous case, we examine the Galerkin, streamline upwind Petrov-Galerkin (SUPG), Galerkin/least-squares (GLS), and adjoint (ADJ) stabilization methods. For the GLS and ADJ methods, we examine formulations constructed from both the "discretize-then-stabilize" technique and the space-time technique. In the spatially discrete case, we examine the Galerkin, least-squares Petrov-Galerkin (LSPG), and adjoint Petrov-Galerkin (APG) methods. We summarize existing analyses for these methods, and provide numerical experiments, which demonstrate that residual-based stabilized methods developed via continuous and discrete processes yield substantial improvements over standard Galerkin methods when the underlying FEM model is under-resolved.
An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection-diffusion-reaction equations. The definition of this scheme contains a new limiter that improves a standard one in such a way that local and global discrete maximum principles are satisfied on arbitrary simplicial meshes.
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz and a corresponding super approximation result (disc…
The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an "almost uniform" error bound (in the sense that the unique term that depends in an unfavorable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robu…
Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is prov…
We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in …
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the app…
Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems
Department of Computer Technologies, Antalya Bilim University（安塔利亚比林大学计算机技术系）；Institute of Applied Mathematics, Middle East Technical University（中东技术大学应用数学研究所）；Department of Mathematics, Indian Institute of Technology Guwahati（印度理工学院古瓦哈提数学系）
We propose and analyze a pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem. The proposed scheme inherits the following advantages from the projection Lagrange--Galerkin scheme. The first advantage is computational efficiency. The scheme decouples the computation of each component of the velocity and pressure. The other advantage is essential unconditional stability. Here we also use the equal-order approximation for the velocity and pressure, and add a symmetric pressure stabilization term. This enriched pressure space enables us to obtain accurate soluti…
We present a Scharfetter-Gummel (SG) stabilization scheme for high-order Hybrid Discontinuous Galerkin (HDG) approximations of convection-diffusion problems. The scheme is based on a careful choice of the stabilization parameters used to define the numerical flux in the HDG method. We show that, in one dimension, the SG-HDG scheme is equivalent to the Finite Volume method stabilized with the Scharfetter--Gummel on the dual grid, for all orders of HDG schemes.
In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that …
We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompr…
This article presents stability and convergence analyses of subgrid multiscale stabilized finite element formulation of non-Newtonian power-law fluid flow model strongly coupled with variable coefficients Advection-Diffusion-Reaction ($VADR$) equation. Considering the highly non-linear viscosity coefficient as solute concentration dependent makes the coupling two way. The stabilized formulation of the transient coupled system is developed based upon time dependent subscales, which ensures inherent consistency of the method. The proposed algebraic expressions of the stabilization parameters app…
This paper considers the discretization of the time-dependent Navier-Stokes equations with the family of inf-sup stabilized Scott-Vogelius pairs recently introduced in [John/Li/Merdon/Rui, arXiv:2206.01242, 2022] for the Stokes problem. Therein, the velocity space is obtained by enriching the H^1-conforming Lagrange element space with some H(div)-conforming Raviart-Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are consi…
A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept
This paper proposes a novel way to solve transient linear, and non-linear solid dynamics for compressible, nearly incompressible, and incompressible material in the updated Lagrangian framework for tetrahedral unstructured finite elements. It consists of a mixed formulation in both displacement and pressure, where the momentum equation of the continuum is complemented with a pressure equation that handles incompresibility inherently. It is obtained through the deviatoric and volumetric split of the stress, that enables us to solve the problem in the incompressible limit. The Varitaional Multi-…
In this paper we present new stability and optimal error analyses of hybridized discontinuous Galerkin (HDG) methods which do not require elliptic regularity assumptions. To obtain error estimates without elliptic regularity assumptions, we use new inf-sup conditions based on stabilized saddle point structures of HDG methods. We show that this approach can be applied to obtain optimal error estimates of HDG methods for the Poisson equations, the convection-reaction-diffusion equations, the Stokes equations, and the Oseen equations.
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure
Institut de Mathématiques de Toulouse, Université Toulouse III & Institut Universitaire de France（图卢兹数学研究所，图卢兹第三大学 & 法国高等研究院）；Institut Camille Jordan, Université Claude Bernard Lyon 1, CNRS UMR 5208（卡米尔·若尔当研究所，里昂第一大学，法国国家科学研究中心）
