Convergence and optimality of the adaptive Morley element method

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem

Convergence of adaptive mixed finite element method for convection-diffusion-reaction equations

We prove the convergence of an adaptive mixed finite element method (AMFEM) for (nonsymmetric) convection-diffusion-reaction equations. The convergence result holds from the cases where convection or reaction is not present to convection-or reaction-dominated problems. A novel technique of analysis is developed without any quasi orthogonality for stress and displacement variables, and without marking the oscillation dependent on discrete solutions and data. We show that AMFEM is a contraction of the error of the stress and displacement variables plus some quantity. Numerical experiments confirm the theoretical results.Comment: arXiv admin note: text overlap with arXiv:1312.645

A Posteriori Error Estimation for the p-curl Problem

We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};\Omega)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It is handled without linearizing around the approximate solution. The non-conformity is dealt by adapting error decomposition techniques of Carstensen, Hu and Orlando. Geometric non-conformities also appear because the continuous problem is defined over a bounded $C^{1,1}$ domain while the discrete problem is formulated over a weaker polyhedral domain. The semi-discrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.Comment: 32 page

Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates

A Posteriori Error Estimates with Computable Upper Bound for the Nonconforming Rotated Q

This paper discusses the nonconforming rotated Q1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

AbstractIn this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf–sup constant is available, which is confirmed by some numerical results

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming CrouzeixRaviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.Facultad de Ciencias Exacta