An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems
In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a saddle-point problem that delivers a stable discrete solution and a robust error estimate that can drive mesh adaptivity. In this work, we demonstrate the efficiency of the method in extreme scenarios, delivering stable solutions. The quality and performance of the solutions are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on each mesh. Meanwhile, this technique allows us to solve on coarse meshes and adapt the solution to achieve a user-specified solution quality.
We develop novel stabilized cut discontinuous Galerkin (CutDG) methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in $\mathbb{R}^d$ where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi)-norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically ro…
A priori analysis for a generalized local projection stabilized finite element approximations for the solution of an advection-reaction equation is presented in this article. The stability and a priori error estimates are established for both the conforming and the nonconforming (Crouzeix-Raviart) approximations with respect to the local projection streamline derivative norm. Finally, the validation of the proposed stabilization scheme and verification of the the derived estimates are presented with appropriate numerical experiments.
We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation .and adaptivity procedure is applied to a series of benchmark problems, demonstrat…
In this paper we have studied subgrid multiscale stabilized formulation with dynamic subscales for non-Newtonian Casson fluid flow model tightly coupled with variable coefficients ADR ($VADR$) equation. The Casson viscosity coefficient is taken to be dependent upon solute mass concentration. This paper presents the stability and convergence analyses of the stabilized finite element solution. The proposed expressions of the stabilization parameters helps in obtaining optimal order of convergences. Appropriate numerical experiments have been provided.
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…
For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of…
We model incompressible flows with an adaptive stabilized finite element method Stokes flows, which solves a discretely stable saddle-point problem to approximate the velocity-pressure pair. Additionally, this saddle-point problem delivers a robust error estimator to guide mesh adaptivity. We analyze the accuracy of different discrete velocity-pressure pairs of continuous finite element spaces, which do not necessarily satisfy the discrete inf-sup condition. We validate the framework's performance with numerical examples.
We devise a stabilized method to weakly enforce bound constraints in the discrete solution of advection-dominated diffusion problems. This method combines a nonlinear penalty formulation with a discontinuous Galerkin-based residual minimization method. We illustrate the efficiency of this scheme for both uniform and adaptive meshes through proper numerical examples.
Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs
We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint …
Boltzmann transport problems often involve heavy streaming, where particles propagate long distance due to the dominance of advection over particle interaction. If an insufficiently refined non-rotationally invariant angular discretisation is used, there are areas of the problem where no particles will propogate. These "ray-effects" are problematic for goal-based error metrics with angular adaptivty, as the metrics in the pre-asymptotic region will be zero/incorrect and angular adaptivity will not occur. In this work we use low-order filtered spherical harmonics, which is rotationally invarian…
In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integ…
When solving time-dependent hyperbolic conservation laws on cut cell meshes one has to overcome the small cell problem: standard explicit time stepping is not stable on small cut cells if the time step is chosen with respect to larger background cells. The domain of dependence (DoD) stabilization is designed to solve this problem in a discontinuous Galerkin framework. It adds a penalty term to the space discretization that restores proper domains of dependency. In this contribution we introduce the DoD stabilization for solving the advection equation in 2d with higher order. We show an $L^2$ s…
We introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the assocociated difficulties is a semi-Lagrangian based reconstruction of subgrid variablity into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a…
In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framew…
We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this residual minimization as a stable saddle-point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection-reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on …
We present stability and error analysis for algebraic flux correction schemes based on monolithic convex limiting. For a continuous finite element discretization of the time-dependent advection equation, we prove global-in-time existence and the worst-case convergence rate of 1/2 w.r.t. the L2 error of the spatial semi-discretization. Moreover, we address the important issue of stabilization for raw antidiffusive fluxes. Our a priori error analysis reveals that their limited counterparts should satisfy a generalized coercivity condition. We introduce a limiter for enforcing this condition in t…
In this paper a fully coupled system of transient $Navier$-$Stokes$ ($NS$) fluid flow model and variable coefficient unsteady Advection-Diffusion-Reaction ($VADR$) transport model has been studied through subgrid multiscale stabilized finite element method. In particular algebraic approach of approximating the subscales has been considered to arrive at stabilized variational formulation of the coupled system. This system is strongly coupled since viscosity of the fluid depends upon the concentration, whose transportation is modelled by $VADR$ equation. Fully implicit schemes have been consider…
In this study, we present a stabilized finite element analysis for completely unified Stokes-Brinkman problems fully coupled with variable coefficient transient Advection-Diffusion-Reaction equation(VADR). As well we have carried out the stabilized finite element analysis for Stokes-Brinkman model with interface conditions fully coupled with VADR. The viscosity of the fluid, involved in flow problem, depends on the concentration of the solute, whose transport is described by VADR equation. The algebraic subgrid multiscale approach has been employed to arrive at the stabilized coupled variation…
Parametric Neural r-Adaptivity for Isogeometric Analysis via Residual Minimization
Universidad del Paı́s Vasco/Euskal Herriko Unibertsitatea (UPV/EHU)（巴斯克大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；Basque Foundation for Science (Ikerbasque)（巴斯克科学基金会）
A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and …
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 tec…
A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Indian Institute of Space Science and Technology（印度空间科学与技术学院）；Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle Ricerche（意大利国家研究委员会应用数学与信息技术研究所）
