In this paper,a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed,based on a type of superconvergence result of the eigenfunction approximation.Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.
In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.
A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.
Aiming at the isoparametric bilinear finite volume element scheme,we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids.Furthermore,we prove that the approximate derivatives are convergent of order two.Finally,numerical examples verify the theoretical results.
In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.
In this paper,we discuss the a posteriori error estimates of the mixed finite element method for quadratic optimal control problems governed by linear parabolic equations.The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions.We derive a posteriori error estimates for both the state and the control approximation.Such estimates,which are apparently not available in the literature,are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.
The ordered patterns formed bymicrophase-separated block copolymer systems demonstrate periodic symmetry,and all periodic structures belong to one of 230 space groups.Based on this fact,a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is developed.In particular,the initial period of the computational box is estimated by the Landau-Brazovskii model as well.By planting the strategy into the whole-space discrete method,several new metastable patterns are discovered in diblock copolymers.
