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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]