A Posteriori Error Estimates of Two-Grid Finite Element Methods for Nonlinear Elliptic Problems

In this article, we study the residual-based a posteriori error estimates of the two-grid finite element methods for the second order nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error in the \(H^1\)-norm are established. Numerical experiments are also provided to illustrate the performance of the proposed estimators.     

Error Estimates and Superconvergence of Mixed Finite Element Methods for Convex Optimal Control Problems

In this paper, we investigate the discretization of general convex optimal control problem using the mixed finite element method. The state and co-state are discretized by the lowest order Raviart-Thomas element and the control is approximated by piecewise constant functions. We derive error estimates for both the control and the state approximation. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problem. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.     

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an $H^1$ -equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectiveness of adaptive mesh refinement guided by this estimator.     

Weighted Error Estimates for Finite Element Solutions of Variational Inequalities

In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes. Received September 6, 1999; revised February 8, 2000     

Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation

In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme.     

Constructive L2 Error Estimates for Finite Element Solutions of the Stokes Equations

Constructive L2 error estimates for finite element solutions of the Stokes equations are described. We show that the L2 error bounds for the velocity can be obtained in a posteriori and explicit a priori sense. Some numerical examples which confirm us the expected rates of convergence are presented.     

Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation

In Grote et al. (SIAM J. Numer. Anal., 44:2408–2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L ²-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation (“leap-frog” scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+Δt ²), where p denotes the polynomial degree, h the mesh size, and Δt the time step.     

Error Estimates for Semidiscrete Finite Element Approximations of the Stokes Equations Under Minimal Regularity Assumptions

Semidiscrete (spatially discrete) finite element approximations of the Stokes equations are studied in this paper. Properties of L ², H ¹ and H ^–1 projections onto discretely divergence-free spaces are discussed and error estimates are derived under minimal regularity assumptions on the solution.     

A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen–Cahn equation u_t − Δu+ ε⁻² f(u)=0. It is shown that the error depends on ε⁻¹ only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen–Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.     

Error Estimates for an LDG Method Applied to Signorini Type Problems

In this paper we propose and analyze a Local Discontinuous Galerkin method for an elliptic variational inequality of the first kind that corresponds to a Poisson equation with Signorini type condition on part of the boundary. The method uses piecewise polynomials of degree one for the field variable and of degree zero or one for the approximation of its gradient. We show optimal convergence for the method and illustrate it with some numerical experiments.     

A Posteriori Error Estimates of Discontinuous Galerkin Method for Nonmonotone Quasi-linear Elliptic Problems

In this paper, we propose and study the residual-based a posteriori error estimates of h-version of symmetric interior penalty discontinuous Galerkin method for solving a class of second order quasi-linear elliptic problems which are of nonmonotone type. Computable upper and lower bounds on the error measured in terms of a natural mesh-dependent energy norm and the broken H ¹-seminorm, respectively, are derived. Numerical experiments are also provided to illustrate the performance of the proposed estimators.     

Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal $L^2$ L 2 -error estimates are derived for semidiscrete approximations, when the initial condition is in $L^2$ L 2 . Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in $L^2,$ L 2 , which improves upon the results available in the literature.     

Star-Based a Posteriori Error Estimates for Elliptic Problems

We give an a posteriori error estimator for low order nonconforming finite element approximations of diffusion-reaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems.     

A Symmetric and Consistent Immersed Finite Element Method for Interface Problems

The non-conforming immersed finite element method (IFEM) developed in Li et al. (Numer Math 96:61–98, 2003) for interface problems is extensively studied in this paper. The non-conforming IFEM is very much like the standard finite element method but with modified basis functions that enforce the natural jump conditions on interface elements. While the non-conforming IFEM is simple and has reasonable accuracy, it is not fully second order accurate due to the discontinuities of the modified basis functions. While the conforming IFEM also developed in Li et al. (Numer Math 96:61–98, 2003) is fully second order accurate, the implementation is more complicated. A new symmetric and consistent IFEM has been developed in this paper. The new method maintains the advantages of the non-conforming IFEM by using the same basis functions but it is symmetric, consistent, and more important, it is second order accurate. The idea is to add some correction terms to the weak form to take into account of the discontinuities in the basis functions. Optimal error estimates are derived for the new symmetric and consistent IFE method in the \(L^2\) and \(H^1\) norms. Numerical examples presented in this paper confirm the theoretical analysis and show that the new developed IFE method has \(O(h^2)\) convergence in the \(L^\infty \) norm as well.