Adaptive Nonconforming Finite Element Methods Research Articles

Pointwise adaptive non-conforming finite element method for the obstacle problem

Pointwise adaptive non-conforming finite element method for the obstacle problem

Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems

Abstract This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m = 1 , 2 m=1,2 . A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.

Convergence of adaptive nonconforming finite element method for Stokes optimal control problems

This paper aims at proving the convergence and quasi-optimality of an adaptive nonconforming finite element method for Stokes distributed control problems with pointwise control constraints. Nonconforming P1/P0 pair (Crouzeix–Raviart elements) and variational discretization are used to approximate the state equation and the control variable, respectively. A posteriori error estimates with upper and lower bounds are first derived for the state and adjoint variables. Then we prove the contraction property for the sum of the energy error of the state and adjoint state and the scaled error estimator on two consecutive adaptive meshes. The resulting linear convergence is finally used to show the quasi-optimal convergence rate of the adaptive algorithm. Additionally, some numerical results are provided to support our theoretical analysis.

On the Convergence of Adaptive Nonconforming Finite Element Methods for a Class of Convex Variational Problems

We formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems. The class of minimization problems we admit includes highly singular problems for which no Euler–Lagrange equation (or inequality) is available. As a consequence, our arguments only use the structure of the energy functional. We are nevertheless able to prove convergence of an adaptive algorithm, using even refinement indicators that are not reliable error indicators.

Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations

We prove convergence and quasi-optimal complexity of adaptive nonconforming low-order finite element methods for the Stokes equations, covering the Crouzeix–Raviart discretization on triangular and tetrahedral meshes, as well as the Rannacher–Turek discretization on two- and three-dimensional rectangular meshes. Hanging nodes are allowed in order to ease local mesh refinement. The adaptive algorithm is based on standard a posteriori error estimators consisting of two parts: a volume residual and an edge term measuring the nonconformity of the velocity approximation. We use an adaptive marking strategies, which, in each step of the iteration, takes only the dominant term into account. This paper can be regarded as an extension of [R. Becker, S. Mao, and Z.-C. Shi, SIAM J. Numer. Anal., 47 (2010), pp. 4639–4659] to the Stokes problem, but the analysis here does not make use of any relationship between mixed and nonconforming finite element methods.

Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems

Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe (in Numer Math 103:251–266, 2006). In this paper, we extend the result to some nonsymmetric and indefinite problems. The main tools in our analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric and indefinite bilinear form. An appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order Crouzeix–Raviart element. It is shown that the ANFEM is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops. Moreover, quasi-optimality in the sense of quasi-optimal algorithmic complexity can be shown for the ANFEM. The results of numerical experiments confirm the theoretical findings.

Convergence of a standard adaptive nonconforming finite element method with optimal complexity

In this paper, we analyze the convergence and optimal complexity of the usual simple adaptive nonconforming finite element method by using Dörfler collective marking strategy. Based on several basic ingredients, such as the estimator reduction, quasi-orthogonality, local upper bound and so on, we eventually show the convergence of the adaptive algorithm by establishing the reduction of some total error and the quasi-optimal convergence rate. Our analysis does not need the relation between the nonconforming P 1 element and the mixed Raviart–Thomas element. The results of numerical experiments confirm that our adaptive algorithm is optimal.

A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity

In this paper, we prove convergence and quasi-optimal complexity of a simple adaptive nonconforming finite element method. In each step of the algorithm, the iterative solution of the discrete system is controlled by an adaptive stopping criterion, and the local refinement is based on either a simple edge residual or a volume term, depending on an adaptive marking strategy. We prove that this marking strategy guarantees a strict reduction of the error, augmented by the volume term and an additional oscillation term, and quasi-optimal complexity of the generated sequence of meshes.

Convergence analysis of an adaptive nonconforming finite element method

An adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations. The analysis is carried out for the lowest order Crouzeix-Raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels. Important tools in the convergence proof are a discrete local efficiency and a quasi-orthogonality property. The proof does neither require regularity of the solution nor uses duality arguments. As a consequence on the data control, no particular mesh design has to be monitored.