Finite Element Analysis. Barna Szabó

Finite Element Analysis

of a discretization scheme, given information about the regularity (smoothness) of the exact solution and (b) a posteriori estimators that provide estimates of the error in energy norm for the finite element solution of a particular problem.

There is a very substantial body of work in the mathematical literature on the a priori estimation of the rate of convergence, given a quantitative measure of the regularity of the exact solution and a sequence of discretizations. The underlying theory is outside of the scope of this book; however, understanding the main results is important for practitioners of finite element analysis. For details we refer to [28, 45, 70, 84].

1.5.1 Regularity

Let us consider problems the exact solution of which has the functional form

(1.89) u Subscript upper E upper X Baseline equals x Superscript alpha Baseline phi left-parenthesis x right-parenthesis comma alpha greater-than 1 slash 2 comma x element-of upper I equals left-parenthesis 0 comma script l right-parenthesis

where phi left-parenthesis x right-parenthesis is an analytic or piecewise analytic function, see Definition A.1 in the appendix. Our motivation for considering functions in this form is that this family of functions models the singular behavior of solutions of linear elliptic boundary value problems near vertices in polygonal and polyhedral domains. For u Subscript upper E upper X to be in the energy space, its first derivative must be square integrable on I. Therefore

integral Subscript 0 Superscript script l Baseline x Superscript 2 left-parenthesis alpha minus 1 right-parenthesis Baseline d x greater-than 0

from which it follows that α must be greater than 1 slash 2 .

In the following we will see that when α is not an integer then the degree of difficulty associated with approximating u Subscript upper E upper X by the finite element method is related to the size of left-parenthesis alpha minus 1 slash 2 right-parenthesis greater-than 0 . The smaller is, the more difficult it is to approximate .

If α is a fractional number then the measure of regularity used in the mathematical literature is the maximum number of square integrable derivatives, with the notion of derivative generalized to fractional numbers. See sections A.2.3 and A.2.4 in the appendix. For our purposes it is sufficient to remember that if u Subscript upper E upper X has the functional form of eq. (1.89), and α is not an integer, then lies in the Sobolev space upper H Superscript alpha plus 1 slash 2 minus epsilon Baseline left-parenthesis upper I right-parenthesis where epsilon greater-than 0 is arbitrarily small. This means that α must be larger than for the first derivative of u Subscript upper E upper X to be square integrable. See, for example, [59].

If α is an integer then u Subscript upper E upper X is an analytic or piecewise analytic function and the measure of regularity is the size of the derivatives of . Analogous definitions apply to two and three dimensions.

Remark 1.9 The kth derivative of a function f left-parenthesis x right-parenthesis is a local property of only when k is an integer. This is not the case for non‐integer derivatives.

1.5.2 A priori estimation of the rate of convergence

Analysts are called upon to choose discretization schemes for particular problems. A sound choice of discretization is based on a priori information on the regularity of the exact solution. If we know that the exact solution lies in Sobolev space upper H Superscript k Baseline left-parenthesis upper I right-parenthesis then it is possible to say how fast the error in energy norm will approach zero as the number of degrees of freedom is increased, given a scheme by which a sequence of discretizations is generated. Index k can be inferred or estimated from the input data κ, c and f.

We define

(1.90) h equals max Underscript j Endscripts script l Subscript j Baseline slash script l comma j equals 1 comma 2 comma ellipsis upper M left-parenthesis normal upper Delta right-parenthesis

where ℓj is the length of the jth element, script l is the size of the of the solution domain upper I equals left-parenthesis 1 comma script l right-parenthesis . This is generalized to two and three dimensions where is the diameter of the domain and ℓj is the diameter of the jth element. In this context diameter means the diameter of the smallest circlein one and two dimensions, or sphere in three dimensions,that contains the element or domain. In two and three dimensions the solution domain is denoted by Ω.

The a priori estimate of the relative error in energy norm for u Subscript upper E upper X Baseline element-of upper H Superscript k Baseline left-parenthesis normal upper Omega right-parenthesis , quasiuniform meshes and polynomial degree p is

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