Condition of finite element matrices generated from nonuniform meshes.

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 ~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