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In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples.

A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete $H^2$ for $k\ge 2$ and in $L^2$ norm for $k>2$ are established for the corresponding weak Galerkin finite element solutions. Numerical results are provided to confirm the theories., arXiv admin note: text overlap with arXiv:1309.5560, arXiv:1510.06001 by other authors

The weak Galerkin form of the finite element method, requiring only C0 basis function, is applied to the biharmonic equation. The computational procedure is thoroughly considered. Local orthogonal bases on triangulations are constructed using various sets of interpolation points with the Gram-Schmidt or Levenberg-Marquardt methods. Comparison and high-precision computations are carried out, and convergence rates are provided up to degree 11 for L2, 10 for H1, and 9 for H2, suggesting that the algorithm is useful for a variety of computations.

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in L∞. The associated equation, coined the ∞-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by Δ2∞u:=(Δu)3|D(Δu)|2=0. In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call ∞-Biharmonic functions. For fixed p we design a mixed finite element scheme for the pre-limiting equation, the p-Bilaplacian Δ2pu:=Δ(|Δu|p−2Δu)=0. We prove convergence of the numerical solution to the weak solution of Δ2pu=0 and show that we are able to pass to the limit p→∞. We perform various tests aimed at understanding the nature of solutions of Δ2∞u and in 1-d we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of -solutions.

Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfurth for the related biharmonic equation. This article generalises Verfurth's result to Kirchhoff plates with an anisotropic material, which requires some modifications. Special emphasis is laid on the reduced Hsieh–Clough–Tocher triangular finite element, the conforming element with the least possible number of unknowns.

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix--Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases.

A new interpolation operator is defined, which preserves only P2 polynomials locally.

In this paper, a $$C^0$$ linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a $$C^0$$ linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under $$L^2$$ and discrete $$H^2$$ norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and $$C^0$$ interior penalty method is given.

We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the $h$-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on different limit surfaces of the extended Loop subdivision for different initial control meshes. Numerical experiments demonstrate that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of $L^2$-norm by theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons., 30 pages, 8 figures, 2 tables

We consider a new three-field formulation of the biharmonic problem. The solution, the gradient and the Lagrange multiplier are the three unknowns in the formulation. Adding a stabilization term in the discrete setting we can use the standard Lagrange finite element to discretize the solution, whereas we use the Raviart-Thomas finite element to discretize the gradient. The Lagrange multipliers are constructed to achieve the optimal error estimate. Numerical results are presented to demonstrate the performance of our approach.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

We analyze a gradient recovery based linear finite element method to solve bi-harmonic equations and the corresponding eigenvalue problems. Our method uses only $$C^0$$C0 element, which avoids complicated construction of $$C^1$$C1 elements and nonconforming elements. Optimal error bounds under various Sobolev norms are established. Moreover, after a post-processing the recovered gradient is superconvergent to the exact one. Some numerical experiments are presented to validate our theoretical findings. As an application, the new method has been also used to solve 1-D fully nonlinear Monge---Ampere equation numerically.

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators. Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.

As modern variant of nonconforming schemes, discontinuous Galerkin finite element meth- ods appear to be highly attractive for fourth-order elliptic PDEs. There exist various modifications and the most prominent versions with first-order convergence properties are the symmetric interior penalty DG method and the C 0 interior penalty method which may compete with the classical Morley nonconforming FEM on triangles. Those schemes differ in their various jump and penalisation terms and also in the norms. This paper proves that the best-approximation errors of all the three schemes are equivalent in the sense that their minimal error in the respective norm and the optimal choice of a discrete approximation can be bounded from below and above by each other. The equivalence constants do only depend on the minimal angle of the triangulation and the penalisation parameter of the schemes; they are independent of any regularity requirement and hold for an arbitrarily coarse mesh.

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method., 28 pages. arXiv admin note: substantial text overlap with arXiv:1303.0927, arXiv:1309.5560, arXiv:1510.06001 by other authors

In this article, we present a method to obtain a C1-surface, defined on a bounded polygonal domain Ω, which interpolates a specific dataset and minimizes a certain “energy functional.” The minimization space chosen is the one associated to the Powell–Sabin finite element, whose elements are C1-quadratic splines. We develop a general theoretical framework for that, and we consider two main applications of the theory. For both of them, we give convergence results, and we present some numerical and graphical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 798–821, 2015

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and ...

A symmetric C 0 finite element method for the biharmonic problem is constructed and analyzed. In our approach, we introduce one-sided discrete second-order derivatives and Hessian matrices to formulate our scheme. We show that the method is stable and converge with optimal order in a variety of norms. A distinctive feature of the method is that the results hold without extrinsic penalization of the gradient across interelement boundaries. Numerical experiments are given that support the theoretical results, and the extension to Kirchhoff plates is also discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1254–1278, 2014

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions., 23 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:1202.3655, arXiv:1204.3655

SUMMARY A goal-oriented algorithm is developed and applied for hp-adaptive approximations given by the discontinuous Galerkin finite element method for the biharmonic equation. The methodology is based on the dual problem associated with the target functional. We consider three error estimators and analyse their properties as basic tools for the design of the hp-adaptive algorithm. To improve adaptation, the combination of two different error estimators is used, each one at its best efficiency, to guide the tasks of where and how to adapt the approximation spaces. The performance of the resulting hp-adaptive schemes is illustrated by numerical experiments for two benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd.

The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate's vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.

In this paper, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $$Q$$ , the finite element space is defined by the span of $$P_2(Q)$$ plus two functions in $$P_3(Q)$$ . Each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them. The degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges. Optimal orders of convergence are proved both in discrete $$H^2$$ and $$H^1$$ seminorms. Several numerical tests confirm the convergence analysis.

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments. © 2006 Elsevier B.V. All rights reserved.

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits natural nested discretization. Then, we present multi-level finite element schemes by implementing the algorithm as in Lin and Xie (Math Comput 84:71–88, 2015) to the nested discretizations on a series of nested grids. The multi-level mixed scheme for the biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

In this article a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable as well as that for the control are order-optimal on general unstructured meshes. However, on uniform meshes not all error estimates are optimal due to the low-order control approximation. All theoretical results are confirmed by numerical tests.

A reliable and efficient residual-based a posteriori error estimator is derived for the Ciarlet-Raviart mixed finite element method for the biharmonic equation on polygonal domains. The performance of the estimator is illustrated by numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

The mixed method for the biharmonic problem introduced in (Behrens and Guzman, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.

In this paper, we consider an optimal control problem governed by elliptic differential equations posed in a three-field formulation. Using the gradient as a new unknown we write a weak equation for the gradient using a Lagrange multiplier. We use a biorthogonal system to discretise the gradient, which leads to a very efficient numerical scheme. A numerical example is presented to demonstrate the convergence of the finite element approach. References D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications. Springer–Verlag, 2013. doi:10.1007/978-3-642-36519-5 . S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer–Verlag, New York, 1994. doi:10.1007/978-0-387-75934-0 . Yanping Chen. Superconvergence of quadratic optimal control problems by triangular mixed finite element methods. International journal for numerical methods in engineering, 75(8):881–898, 2008. doi:10.1002/nme.2272 . Hongfei Fu, Hongxing Rui, Jian Hou, and Haihong Li. A stabilized mixed finite element method for elliptic optimal control problems. Journal of Scientific Computing, 66(3):968–986, 2016. doi:10.1007/s10915-015-0050-3 . Hui Guo, Hongfei Fu, and Jiansong Zhang. A splitting positive definite mixed finite element method for elliptic optimal control problem. Applied Mathematics and Computation, 219(24):11178–11190, August 2013. doi:10.1016/j.amc.2013.05.020 . Muhammad Ilyas and Bishnu P. Lamichhane. A stabilised mixed finite element method for the poisson problem based on a three-field formulation. In M. Nelson, D. Mallet, B. Pincombe, and J. Bunder, editors, Proceedings of EMAC-2015, volume 57 of ANZIAM J., pages C177–C192. Cambridge University Press, 2016. doi:10.21914/anziamj.v57i0.10356 . Bishnu P Lamichhane, AT McBride, and BD Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Computer Methods in Applied Mechanics and Engineering, 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008 . B.P. Lamichhane. Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity. IMA Journal of Numerical Analysis, 29:404–420, 2009. doi:10.1093/imanum/drn013 . B.P. Lamichhane. A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. Journal of Scientific Computing, 46:379–396, 2011. doi:10.1007/s10915-010-9409-7 . B.P. Lamichhane and E. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numerical Methods for Partial Differential Equations, 28:1336–1353, 2012. doi:10.1002/num.20683 . Xianbing Luo, Yanping Chen, and Yunqing Huang. Some error estimates of finite volume element approximation for elliptic optimal control problems. International Journal of Numerical Analysis and Modeling, 10(3):697–711, 2013. http://www.math.ualberta.ca/ijnam/Volume-10-2013/No-3-13/2013-03-11.pdf . Fredi Troltzsch. On finite element error estimates for optimal control problems with elliptic PDEs. In International Conference on Large-Scale Scientific Computing, pages 40–53. Springer, 2009. doi:10.1007/978-3-642-\(12535-5_4\) . Fredi Troltzsch. Optimal control of partial differential equations, volume 112. American Mathematical Society, 2010. http://www.ams.org/books/gsm/112/ .

In this article, we propose a new family of high regularity finite element spaces. The global approximation spaces are obtained in two steps. We first build an open cover of the computational domain and local approximation spaces on each patch of the cover. Then we construct partition of unity functions subordinate to the open cover depending on the regularity requirement. The basis functions of the global space is given by the products of the local basis functions and the corresponding partition of unity functions. The method can be used to construct finite element spaces of any desired regularity. Approximation properties and implementation details are discussed. Numerical examples for the biharmonic equation are presented to show the effectiveness of the proposed method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 1–16, 2012

This paper introduces and analyses a local, residual-based a posteriori error indicator for the Morley finite element method of the biharmonic Kirchhoff plate bending problem. In the theoretical part of the paper, a recent approach presented by the authors for clamped boundaries is extended to general boundary conditions. The error indicator is proven to be both reliable and efficient. The numerical part of the paper presents a set of results on various benchmark computations with different kinds of domains and boundary conditions. These tests verify the reliability and efficiency of the error estimator and illustrate the robustness of the method for adaptive mesh refinements. Copyright © 2009 John Wiley & Sons, Ltd.

A penalty method for mixed finite element methods is formulated and studied. The Herrmann-Miyoshi scheme for the biharmonic equation is considered. The main idea is to build a perturbed problem with two parameters playing the role of penalties. The perturbed problem is constructed by replacing principal conditions in the mixed variational formulation at the interface by natural conditions containing parameters. Discretization of the perturbed problem is effected by a finite element method. Estimates for the norm of the difference between the solutions of a discrete perturbed problem and of an initial value problem are derived depending on the mesh size and penalties. Recommendations are given as to how to choose penalties so as to fit a mesh size.

In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation. We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfurth (A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three dimensional case.

A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

In this paper, the W 2 , p -error and Lp-error estimates of conforming finite element methods for biharmonic problem are established via weight function technique, where 2 ⩽ p ⩽ +∞. Finally, the numerical experiments are provided.

In this paper, we apply the effective condition number in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83] to the penalty Trefftz methods (TMs) with high order elements such as Adini's elements for Poisson's equations with singularities. The best superconvergence rates O ( h 3.5 ) in H 1 norm can be reached, where h is the maximal boundary length of elements. For the penalty and the penalty plus hybrid Trefftz methods, the bounds of effective condition numbers retain the same. However, for the penalty TM combination, not only are the algorithms simple, but also the numerical solutions are more accurate. Hence we prefer the penalty TM combination to the penalty plus hybrid TM combination. Such a recommendation is based on the effective condition number, but not on the traditional condition number, based on which the penalty TM combination was declined in Li and Yan [Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs for biharmonic equations. Appl Numer Math 2001;39:61–85]. The new stability analysis in this paper by means of effective condition number may re-evaluate the existing numerical methods; this paper reports the results of Motz's problem. Note that in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83], no stability analysis was provided.