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The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∊ V := H2 0 (\Omega) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces Vh(P) and a smoother allows rough source terms F ∊ V* =H-20(Ω). The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator J: Vh → V from the nonconforming virtual element space Vh. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on u ∊ V. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [ABSTRACT FROM AUTHOR]

We prove an optimal-order error bound in the discrete H2() norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in n space dimensions, with n 2 f2; : : : ; 7g, whose generalized solution belongs to the Sobolev space Hs()\H2 0 () for 1 2 max(5; n) < s 4, where = (0; 1)n. The result extends the range of the Sobolev index s in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of Hs() into C() in n space dimensions. [ABSTRACT FROM AUTHOR]

For biharmonic boundary value problems, the Ciarlet-Raviart mixed method is considered on polygonal domains without additional convexity assumptions. Mapping properties of the involved operators on the continuous as well as on the discrete level is studied. Based on this, efficient preconditioners are constructed and numerical experiments are shown. [ABSTRACT FROM AUTHOR]

We develop a fast Fourier-Galerkin method for solving a boundary integral equation which is a reformulation of the Dirichlet problem of the biharmonic equation. The proposed method is based on a splitting of the resulting boundary integral operator. That is, we write the operator as a sum of two integral operators, one having the Fourier basis functions as eigenfunctions and the other whose representation matrix in the Fourier basis can be compressed to a sparse matrix having only O(n log n) number of nonzero entries, where n is the order of the Fourier basis functions used in the method. We then project the solution of the boundary integral equation onto the space spanned by the Fourier basis functions. This leads to a system of linear equations. A fast solver for the system is based on a compression of its coefficient matrix. We show that the method has the optimal convergence order O(n-q), where q denotes the degree of regularity of the exact solution of the boundary integral equation and requires computing only O(n log n) number of entries of the coefficient matrix. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method. [ABSTRACT FROM AUTHOR]

A macro type of biquadratic C¹ finite elements is constructed on rectangle grids. This is a rectangular version of the C¹ Powell-Sabin element, a C¹-P2 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 C¹-Q2 space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional C¹-P2 spaces, where the nodal basis is supported on four intervals. The B-spline function of P2 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. [ABSTRACT FROM AUTHOR]

We discuss a generalization of the Cauchy­Riemann 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 L2(Ω) are proved in regions with smooth boundaries and optimal rates of convergence are proved in H1(Ω) and, using linear triangular elements, observed in L2(Ω) in polygonal regions. [ABSTRACT FROM AUTHOR]

We present a new Lagrange multiplier-based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the Hsieh­Clough­Tocher element, the discrete Kirchhoff triangle, and a class of nonlocking elements for the Reissner­Mindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. Computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64-processor IBM SP2 of a plate bending problem with almost one million degrees of freedom. [ABSTRACT FROM AUTHOR]