Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.     

Exact integration formulas for the finite volume element method on simplicial meshes

This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high‐order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)—Gaussians, sech's and Inverse Quadratics—the condition number κ(α,N) rapidly asymptotes to a limit κasymp(α) that is independent of N and depends only on α, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed α grows as N². For all four, there is growth proportional to an exponential of 1/α (1/α² for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N—a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.     

Numerical simulation of the consequences of a mechanical action on a human brain under a skull injury

A two-dimensional mathematical model of the mechanical response of a human head to a shock action is proposed. It describes the spatial distribution of the mechanical loads on the brain. Some numerical results obtained using the grid characteristic methods on unstructured triangular grids are presented.     

The finite volume method based on the Crouzeix–Raviart element for a fracture model

In this paper, we present and analyze a finite volume method based on the Crouzeix–Raviart element for the coupled fracture model, where the fluid flow is governed by Darcy's law in the one‐dimensional fracture and two‐dimensional surrounding matrix. In the numerical scheme, the pressure in the matrix and fracture is respectively approximated by the Crouzeix–Raviart elements and piecewise constant functions, and then the velocity is calculated by piecewise constant functions element by element. The existence and uniqueness of the numerical solution are discussed, and optimal order error estimates for both the pressure p and the velocity u are proved on general triangulations. We finally carry out numerical experiments, and results confirm our theoretical analysis.     

Technical Note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes

We obtain a computable lower bound on the value of the interior penalty parameters sufficient for the existence of a unique discontinuous Galerkin finite element approximation of a second order elliptic problem. The bound obtained is valid for meshes containing an arbitrary number of hanging nodes and elements of arbitrary nonuniform polynomial order. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

We prove the homogenisation to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. Due to the incompressibility condition and the lack of a maximum principle for the Stokes equations, our proof requires a very careful study of the geometry of the random holes generated by the class of probability measures considered.     

Symmetries and similarity solutions for the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane

We perform a complete analysis of all the Lie point symmetries admitted by the equation describing the axisymmetric spreading under gravity of a thin power-law liquid drop on a horizontal plane. We then investigate the existence of group-invariant solutions for particular values of the power-law parameter β.     

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt) . Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.