Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Condition of finite element matrices generated from nonuniform meshes.

Abstract: N a previous Note1 it has been shown (see also Refs. 2 and 3) that the spectral condition number Cn(K) of the global (stiffness) matrix K arising from a uniform mesh of finite elements (or of finite differences) discretization can be expressed by Cn(K) = cNes2m where 2m is the order of the differential equation and c a coefficient independent of Nes, the number of elements per side, but dependent on the order of the interpolation polynomials inside the element. This condition is "natural", since it is inherently associated with the approximation of the continuous problem by the discrete (algebraic) one. Nonuniform meshes of finite elements introduce many additional factors which may adversely affect the condition of the system. It is the purpose of this Note to describe a technique for establishing bounds on the condition number for irregular meshes of finite elements. With these bounds it becomes possible to study the effect of element geometry, the order of interpolation functions and other intrinsic and discretization parameters on Cn(K) and to isolate the factors that may lead to ill-conditioning. The matrix K is termed ill-conditioned when \Q~sCn(K) = 1, where s denotes the number of decimals in the computer. The bounds on Cn(K) are expressed in terms of the extremal eigenvalues of the element matrices. Since the element matrices are of restricted size, derivation of the bounds on Cn(K) as a function of the discretization parameters become straightforward for any problem and any element. Particular attention is focused on the possibility of improving the condition of the matrix by scaling. Bounds on the Extremal Eigenvalues

Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations

Abstract: Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the schemes of the Hybrid Mimetic Mixed family, which include in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above-mentioned problems.

Full-vectorial mode calculations by finite difference method

Abstract: Finite difference calculations of full-vectorial modes of optical waveguides are presented. This method has overcome the limitations of the semivectorial approximation and is able to calculate full-vectorial modes of arbitrary order for a given structure with an arbitrary refractive index profile. Numerical results show that the method is accurate. In addition, the leakage phenomenon of the higher modes of a rib waveguide is predicted by this method.