360 research outputs found
A Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients
In this paper, we develop a gradient recovery based linear (GRBL) finite
element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second
order elliptic equations in non-divergence form. The elliptic equation is
casted into a symmetric non-divergence weak formulation, in which second order
derivatives of the unknown function are involved. We use gradient and Hessian
recovery operators to calculate the second order derivatives of linear finite
element approximations. Although, thanks to low degrees of freedom (DOF) of
linear elements, the implementation of the proposed schemes is easy and
straightforward, the performances of the methods are competitive. The unique
solvability and the seminorm error estimate of the GRBL scheme are
rigorously proved. Optimal error estimates in both the norm and the
seminorm have been proved when the coefficient is diagonal, which have been
confirmed by numerical experiments. Superconvergence in errors has also been
observed. Moreover, our methods can handle computational domains with curved
boundaries without loss of accuracy from approximation of boundaries. Finally,
the proposed numerical methods have been successfully applied to solve fully
nonlinear Monge-Amp\`{e}re equations
Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media
A conservative flux postprocessing algorithm is presented for both
steady-state and dynamic flow models. The postprocessed flux is shown to have
the same convergence order as the original flux. An arbitrary flux
approximation is projected into a conservative subspace by adding a piecewise
constant correction that is minimized in a weighted norm. The application
of a weighted norm appears to yield better results for heterogeneous media than
the standard norm which has been considered in earlier works. We also
study the effect of different flux calculations on the domain boundary. In
particular we consider the continuous Galerkin finite element method for
solving Darcy flow and couple it with a discontinuous Galerkin finite element
method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table
Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials
We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial-scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning
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