We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.
The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a~priori error analysis and an a~posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data and so has to provide discrete stability by a perturbation analysis be…
This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of nonconformity errors. A reconstruction formula provides the discrete solution of the dual problem via a simple postprocessing procedure which implies a strong duality relation and is of interest in a posteriori error estimation. The framework applies to differentiable and nonsmooth problems, examples include $p$-Laplace, total-variation regularized, and obstacle probl…
We propose in this paper a discretization of the momentum convection operator for fluid flow simulations on quadrangular or hexahedral meshes. The space discretization is performed by the loworder nonconforming Rannacher-Turek finite element: the scalar unknowns are associated to the cells of the mesh, while the velocities unknowns are associated to the edges or faces. The momentum convection operator is of finite volume type, and its almost second order expression is derived by a MUSCL-like technique. The latter is of algebraic type, in the sense that the limitation procedure does not invoke …
The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\in V:=H^2_0(Ω)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\in V^*=H^{-2}(Ω)$. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator $J:V_h\to V$ from the nonconforming virtual element space $V_h$. The operator $J$ is a right-inve…
Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix-- Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties, the first one is the construction of high accuracy gradient recovery algorithms, the second one is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are …
$\boldsymbol{H}(\textbf{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations
Inria（法国国家信息与自动化研究所）；CERMICS, ENPC, Institut Polytechnique de Paris, CNRS（巴黎高等土木工程学校、巴黎理工学院、法国国家科学研究中心计算数学与计算机科学中心）
The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space $H^1_0(Ω)$ by a right-inverse $J$ of the interpolation operator $I_h$. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for opti…
We present a robust a posteriori error estimator for the weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The estimator provides global upper and lower bounds of the error %measured in a suitable norm and is robust in the sense that upper and lower bounds are uniformly bounded with respect to the diffusion coefficient. Results of the numerical experiments are presented to illustrate the performance of the error estimator.
We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For convection-dominated problems, the norm introduced by Verf{ü}rth \cite{verf2005confusion} is used to measure the approximation error. Various numerical experiments are performed to (1) demonstrate the robustness of the hybrid estimator; (2) show that the hybri…
In this work we derive a posteriori error estimates for the convection-diffusion-reaction equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending on the concentration of the fluid. We introduce the variational formulation associated to the problem, and discretize it by using the finite element method. We prove optimal a posteriori errors with two types of calculable error indicators. The first one is linked to the linearization and the second one to the discretization. Then we find upper and lower error bounds under additional regularity assumptions on the …
This article on nonconforming schemes for $m$ harmonic problems simultaneously treats the Crouzeix-Raviart ($m=1$) and the Morley finite elements ($m=2$) for the original and for modified right-hand side $F$ in the dual space $V^*:=H^{-m}(Ω)$ to the energy space $V:=H^{m}_0(Ω)$. The smoother $J:V_{\rm nc} \to V$ in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator $I_{\rm nc} :V\to V_{\rm nc}$, and modifies the discrete right-hand side $F_h:=F\circ J \in V_{\rm nc}^*$. The best-approximation property of the modified schem…
We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions [A. Abdulle and G.Rosilho de Souza, ESAIM: M2AN, 53(4):1269--1303, 2019]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy…
A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
Université de Lille 1（里尔第一大学）；INRIA Lille Nord Europe（法国国家信息与自动化研究所里尔北部分支）；Université de Valenciennes et du Hainaut Cambrésis（瓦朗谢纳及康布雷埃斯海诺特大学）
Error Analysis for a Nonconforming Finite Element Discretization of the 3D-CCPF Model
Institut de Mathématiques et de Sciences Physiques (IMSP)（数学与物理科学研究所）；Faculté des Sciences et Techniques (FAST) Université d’Abomey-Calavi (UAC)（阿波美-卡拉维大学科学技术学院）
A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept
Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error of two discrete finite element solutions based on two nested triangulations. In the error analysis of nonconforming finite element methods, like the Crouzeix-Raviart or Morley finite element schemes, the difference of the piecewise derivatives of discontinuous approximations to the distributional gradients of global Sobolev functions plays a dominant role a…
This article discusses numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in $\mathbb{R}^2$. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.
Two residual-type error estimators for the mortar staggered discontinuous Galerkin discretizations of second order elliptic equations are developed. Both error estimators are proved to be reliable and efficient. Key to the derivation of the error estimator in potential $L^2$ error is the duality argument. On the other hand, an auxiliary function is defined, making it capable of decomposing the energy error into conforming part and nonconforming part, which can be combined with the well-known Scott-Zhang local quasi-interpolation operator and the mortar discrete formulation yields an error esti…
We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the {convective derivative of the} error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error which functions as a maximum-norm counterpart to the…
An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations
On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems
The Pennsylvania State University（宾夕法尼亚州立大学）；Rutgers, The State University of New Jersey（新泽西州立罗格斯大学）；Institute of Computational Mathematics, Chinese Academy of Sciences（中国科学院计算数学与科学工程计算研究所）
Pressure-robust $hp$-a posteriori error estimates of $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin methods for the Stokes equations
Inria（法国国家数字技术研究院）；CERMICS, CNRS, ENPC, Institut Polytechnique de Paris（巴黎高等师范学院、国立信息与自动化研究所、巴黎理工学院）；School of Mathematical Sciences（东方理工学院数学科学学院）；Eastern Institute of Technology（香港城市大学数学系）；Department of Mathematics, City University of Hong Kong
