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International audience; We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

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

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.

This paper is devoted to the $L^2$ norm error estimate of the Adini element for the biharmonic equation. Surprisingly, a lower bound is established which proves that the $L^2$ norm convergence rate cannot be higher than that in the energy norm. This proves the conjecture of [Lascaux and Lesaint, RAIRO Anal. Numer., 9 (1975), pp. 9--53] that the convergence rates in both $L^2$ and $H^1$ norms cannot be higher than that in the energy norm for this element.

A macro type of biquadratic $C^1$ finite elements is constructed on rectangle grids. This is a rectangular version of the $C^1$ Powell-Sabin element, a $C^1$-$P_2$ element on triangular grids. Here, each rectangle of the base grid is refined into four subrectangles. As in the case of the Powell-Sabin element, we have more constraints than the number of degrees of freedom on each macroelement. However, the extra constraints are consistent. It is shown further that the constructed finite element space is the full $C^1$-$Q_2$ space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional $C^1$-$P_2$ spaces, where the nodal basis is supported on four intervals. The B-spline function of $P_2$ is supported on three intervals. The Girault-Scott operator is extended to the element. The application and the convergence of the finite element to the biharmonic equation are presented. Numerical tests are provided.

We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate $\Delta^{2}\Phi$ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is $\Delta^{2}Q_{\Phi}(0,0)$, where $Q_{\Phi}$ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303-333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.

This paper is devoted to the analysis of a new compact scheme for the Navier-Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. Ben-Artzi e, J. Comput. Phys., 205 (2005), pp. 640-664]. The scheme discussed here combines the Stephenson scheme for the biharmonic operator and ideas from box-scheme methodology. Consistency and convergence are proved for the full nonlinear system. Instead of customary periodic conditions, the case of boundary conditions is addressed. It is shown that in one dimension the truncation error for the biharmonic operator is $O(h^4)$ at interior points and $O(h)$ at near-boundary points. In two dimensions the truncation error is $O(h^2)$ at interior points (due to the cross-terms) and $O(h)$ at near- boundary points. Hence the scheme is globally of order four in the one-dimensional periodic case and of order two in the two-dimensional periodic case, but of order 3/2 for one- and two- dimensional nonperiodic boundary conditions. We emphasize in particular that there is no special treatment of the boundary, thus allowing robust use of the scheme. The finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications.

A new smoothing method is proposed which can be viewed as a finite element thin plate spline. This approach combines the favorable properties of finite element surface fitting with those of thin plate splines. The method is based on first order techniques similar to mixed finite element techniques for the biharmonic equation. The existence of a solution to our smoothing problem is demonstrated, and the approximation theory for uniformly spread data is presented in the case of both exact and noisy data. This convergence analysis seems to be the first for a discrete smoothing spline with data perturbed by white noise. Numerical results are presented which verify our theoretical results and demonstrate our method on a large real life data set.

In this work we present a finite element method for the biharmonic problem based on the primal mixed formulation of Ciarlet and Raviart [A mixed finite element method for the biharmonic equation, in Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 125--143]. We introduce a dual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliary variable with no computational effort. Thus, the discrete problem turns to be governed by a system of linear equations with symmetric and positive definite coefficients and can be solved by classical algorithms. The construction of the stiffness matrix is obtained by using Courant triangles and can be done with great efficiency.

We discuss a generalization of the Cauchy--Riemann equations, which is applicable to biharmonic problems. A first-order system is derived from these equations, and this system is solved using the least squares method with continuous trial functions set up by the finite element method. Optimal rates of convergence in $L^2 (\Omega)$ are proved in regions with smooth boundaries and optimal rates of convergence are proved in $H^1 (\Omega)$ and, using linear triangular elements, observed in $L^2 (\Omega)$ in polygonal regions.

Additive Schwarz methods are overlapping domain decomposition methods proposed by Dryja and Widlund for second-order elliptic problems. In this paper, we consider two-level additive Schwarz methods for the biharmonic Dirichlet problem discretized by conforming $C^1 $ finite elements. Most of these elements are nonnested in the sense that the finite element space defined on the coarse mesh is not a subspace of the space defined on the finer mesh. We construct certain intergrid transfer operators and establish that the algorithms have optimal convergence properties. Our algorithms include the cases when the two-level triangulation are nonnested and the subspaces on the coarse and the fine grids are defined by different finite elements. Our analysis is based on the theory of Dryja and Widlund and our estimates use the stable approximation properties of the finite elements and the intergrid transfer operators.

An integral equation method for the Dirichlet problem for the biharmonic equation is proposed. It leads to a $2 \times 2$ matrix integral equation system. By taking suitable norms on the spaces of density functions, the Fredholm operator theory can be used to prove the solvability. The kernels in this system are relatively complicated. Therefore, especially when a high-order polynomial approximation is used for a numerical purpose, it is costly to evaluate the integrals that appear in the numerical system. A discrete Galerkin method that has shown superb convergence is proposed here, as in [K. Atkinson, J. Integral Equations Appl., 1 (1988), pp. 343–363] and elsewhere. When the boundary functions are smooth, exponential convergence is observed.

A multigrid preconditioning scheme for solving the Ciarlet–Raviart mixed method equations for the biharmonic Dirichlet problem is presented. In particular, a Schur complement formulation for these equations which yields noninherited quadratic forms is considered. The preconditioning scheme is compared with a standard W-cycle multigrid iteration. It is proved that a Variable V-cycle preconditioner leads to problems with uniformly bounded condition numbers. However, W-cycle convergence is proved only if the number of smoothings “m is sufficiently large.” An example is given in which the W-cycle diverges unless $m \geq 8$. Divergent W-cycles are also encountered when solving the Morley equations for the biharmonic Dirichlet problem, although Brenner has proved W-cycle convergence for sufficiently large m [9]. This is illustrated with additional computations, while Variable V-cycles continue to produce excellent preconditioners in this setting.Certain approximate $L_2$-inner products are described and a modif...

The paper deals with hierarchical bases in spaces of conforming $C^1 $ elements in connection with the approximate solution of the biharmonic equation \[ \Delta ^2 u = f\quad {\text{in }}\Omega ,\qquad u = \frac{{\partial u}}{{\partial n}} = 0\quad {\text{on }}\partial \Omega \]on a plane polygonal domain $\Omega $. Two different composite finite elements are studied: piecewise quadratic Powell–Sabin elements and piecewise cubic elements of Clough–Tocher type.The main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization (the so-called hierarchical multilevel method) will yield suboptimal convergence rates in comparison with standard multigrid schemes.

Modifications of the Morley method for the approximation of the biharmonic equation are obtained from various finite element methods applied to the equations of linear isotropic elasticity and the stationary Stokes equations, by elimination procedures analogous to those used in the continuous case. Problems with Korn’s first inequality for nonconforming $P_1 $ elements and its implications for the approximation of the elasticity equations are also discussed.

A new method for the numerical solution of the first biharmonic Dirichlet problem in a rectangular domain is presented. For an $N \times N$ mesh the complexity of this algorithm is on the order of ...

In the p-version of the finite element method, the triangulation of the domain is fixed and the degree p of the local approximating polynomial is increased. This is in contrast with the h-version in which the degree p is fixed and the maximum diameter h of all triangles is allowed to go to zero. We derive a convergence result for the approximating solution to the biharmonic equation when using the p-version. This result shows that the rate of convergence for the p-version is twice that of h-version with quasi-uniform mesh in the approximation of singular solutions caused by corners. This result which applies to problems which require $C^1$ global continuity is analogous to an earlier result which applied to $C^0$ problems.Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared with theoretical predictions.

The Cahn–Hilliard equation is a nonlinear evolutionary equation that is fourth order in space. In this paper a continuous in-time finite-element Galerkin approximation is considered. We use the nonconforming Morley element and derive optimal order error bounds in $L^2 $.

The biharmonic boundary value problem with Dirichlet boundary conditions is reduced to a coupled system of Poisson equations, which depends upon an arbitrary, positive coupling constant c. Since each of the Poisson equations is well-posed, the system may be solved by iteration. We show that the iterates may be represented as a linear combination of the eigenfunctions of the “Dirichlet eigenvalue problem” (a fourth order boundary value problem with the eigenvalue in the boundary condition). Convergence of the iterative scheme occurs when $0 < c < 2v_1$, where $v_1 $ is the smallest eigenvalue. By making use of an averaging scheme, convergence may be produced for any positive c. With the proper choice of c, the rate of convergence may be increased. This coupled equation approach includes the finite difference approach as a special case.

“Optimal” order of convergence estimates are derived for a new mixed element approximation for the biharmonic problem.

We present fast methods for solving Laplace’s and the biharmonic equations on irregular regions with smooth boundaries. The methods used for solving both equations make use of fast Poisson solvers on a rectangular region in which the irregular region is embedded. They also both use an integral equation formulation of the problem where the integral equations are Fredholm integral equations of the second kind. The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region. Fast solvers are then used to compute the extended solution. Aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth.

The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.

We consider the mixed finite element approximation of the biharmonic problem on a smooth domain with either clamped or simply supported boundary conditions. Error estimates are proved for the Ciarlet–Raviart formulation of the clamped plate problem and for a similar method for the simply supported plate problem.

This paper considers the Dirichlet problem for the two-dimensional biharmonic equation in a bounded region consisting of a finite sum of rectangles. The biharmonic equation is first split into two Poisson equations and two classes of finite difference schemes are defined for obtaining the numerical solution. These classes correspond to the type of difference approximation defined for the missing boundary condition. Discretization error for the difference schemes in these two classes is shown to be of order $h^{{3 / 2}} $ and $h^2 $, respectively, as the mesh size $h \to 0$. The effect on discretization error of the different approximations within a class is also examined.

The discrete biharmonic equation on a rectangular region and the discrete Poisson equation on an irregular region can be treated as modifications to matrix problems with very special structure. We show how to use the direct method of matrix decomposition to formulate an effective numerical algorithm for these problems. For typical applications the operation count is $O(N^3 )$ for an $N \times N$ grid. Numerical comparisons with other techniques are included.

Biharmonic functions are much used in the theory of elasticity. A majorization formula gives an upper bound for $|u|$ in terms of the boundary conditions of u, where u is a biharmonic function. Such a formula would be useful in estimating the size of u, providing one has explicit constants in the formula. Apparently the best results to date (C. Miranda, Formule di maggiorazione a teorema di esistenza per le funzioni biarmoniche di due variabili, Giorn. Mat. Battaglini, 78 (1948–49), pp. 97–118) had shown that a formula exists for each region, but gave no way to calculate the constants appearing in the formula for any region. In this report are supplied explicit constants for the two cases of a circle and a rectangle.

A priori inequalities are used as the basis of a method for obtaining approximate solutions (with norm and pointwise error bounds) to the first biharmonic boundary value problem. The approximations are expressed in terms of trial functions which need not satisfy either the differential equation or the boundary conditions. Very accurate approximations were found in the numerical examples treated.

Coupled equation approach to finite difference solution of biharmonic equation, noting outer iteration scheme for pair of discrete Poisson equations

A technique is proposed for solving the finite difference biharmonic equation as a coupled pair of harmonic difference equations. Essentially, the method is a general block SOR method with convergence rate $O(h^{{1 / 2}} )$ on a square, where h is mesh size.