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From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

In this paper, numerical approximation for two-dimensional (2D) multiterm time fractional diffusion equation is considered. By virtue of properties of bilinear element, Raviart--Thomas element and L1 approximation, an H¹-Galerkin mixed finite element fully discrete approximate scheme is established for 2D multi-term time fractional diffusion equation. And then, unconditionally stable of the approximate scheme is rigourously testified by dealing with fractional derivative skilfully. At the same time, superclose results for the original variable u in H¹-norm and the flux q = ∇u in H(div, Ω)-norm are derived. Furthermore, the global superconvergence results for u in H¹-norm are deduced by the interpolation postprocessing operator. Finally, numerical results demonstrate that the approximate scheme provides a valid and efficient way for solving 2D multi-term time fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators. [ABSTRACT FROM AUTHOR]

Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the approximation ofp-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of thep-Laplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations. [ABSTRACT FROM AUTHOR]

In this paper, a parallel Robin–Robin domain decomposition method for H(div)-elliptic problems is proposed. The convergence of the method is proved for both the continuous problem and the finite element approximation. Some numerical testes are also presented to demonstrate the effectiveness of the method. [ABSTRACT FROM PUBLISHER]