Finite Element Analysis. Barna Szabó

Finite Element Analysis

StartFraction h Superscript k minus 1 Baseline Over p Superscript k minus 1 Baseline EndFraction double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper H Sub Superscript k Subscript left-parenthesis normal upper Omega right-parenthesis Baseline 2nd Column for k minus 1 less-than-or-equal-to p 2nd Row 1st Column Blank 2nd Column Blank 3rd Row 1st Column upper C left-parenthesis k right-parenthesis StartFraction h Superscript p Baseline Over p Superscript k minus 1 Baseline EndFraction double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper H Sub Superscript p plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline 2nd Column for k minus 1 greater-than p EndMatrix"/>

where upper E left-parenthesis normal upper Omega right-parenthesis is the energy norm, k is typically a fractional number and upper C left-parenthesis k right-parenthesis is a positive constant that depends on k but not on h or p. This inequality gives the upper bound for the asymptotic rate of convergence of the relative error in energy norm as h right-arrow 0 or p right-arrow infinity [22]. This estimate holds for one, two and three dimensions. For one and two dimensions lower bounds were proven in [13, 24] and [46] and it was shown that when singularities are located in vertex points then the rate of convergence of the p‐version is twice the rate of convergence of the h‐version when both are expressed in terms of the number of degrees of freedom. It is reasonable to assume that analogous results can be proven for three dimensions; however, no proofs are available at present.

We will find it convenient to write the relative error in energy norm in the following form

(1.92) left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E Baseline less-than-or-equal-to StartFraction upper C Over upper N Superscript beta Baseline EndFraction

where N is the number of degrees of freedom and C and β are positive constants, β is called the algebraic rate of convergence. In one dimension upper N proportional-to 1 slash h for the h‐version and upper N proportional-to p for the p‐version. Therefore for k minus 1 less-than p we have beta equals k minus 1 . However, for the important special case when the solution has the functional form of eq. (1.89) or, more generally, has a term like u equals StartAbsoluteValue x minus x 0 EndAbsoluteValue Superscript lamda and x 0 element-of upper I overbar is a nodal point then beta equals 2 left-parenthesis k minus 1 right-parenthesis for the p‐version: The rate of p‐convergence is twice that of h‐convergence [22, 84].

When the exact solution is an analytic function then u Subscript upper E upper X Baseline element-of upper H Superscript infinity Baseline left-parenthesis normal upper Omega right-parenthesis and the asymptotic rate of convergence is exponential:

(1.93) left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E Baseline less-than-or-equal-to StartFraction upper C Over exp left-parenthesis gamma upper N Superscript theta Baseline right-parenthesis EndFraction

where C, γ and θ are positive constants, independent of N. In one dimension theta greater-than-or-equal-to 1 slash 2 , in two dimensions , in three dimensions , see [10].

When the exact solution is a piecewise analytic function then eq. (1.93) still holds provided that the boundary points of analytic functions are nodal points, or more generally, lie on the boundaries of finite elements.

The relationship between the error e equals u Subscript upper E upper X Baseline minus u Subscript upper F upper E measured in energy norm and the error in potential energy is established by the following theorem.

Theorem 1.5

(1.94)

Proof: Writing e equals u Subscript upper E upper X Baseline minus u Subscript upper F upper E and noting that e element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis , from the definition of pi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis we have:

StartLayout 1st Row 1st Column pi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis equals 2nd Column pi left-parenthesis u Subscript upper E upper X Baseline minus e right-parenthesis equals one half upper B left-parenthesis u Subscript upper E upper X Baseline minus e comma u Subscript upper E upper X Baseline minus e right-parenthesis minus upper F left-parenthesis u Subscript upper E upper X Baseline minus e right-parenthesis 2nd Row 1st Column equals 2nd Column one half upper B left-parenthesis u Subscript upper E upper X Baseline comma u Subscript upper E upper X Baseline right-parenthesis minus upper F left-parenthesis u Subscript upper E upper X Baseline right-parenthesis ModifyingBelow minus upper B left-parenthesis u Subscript upper E upper X Baseline comma e right-parenthesis plus upper F left-parenthesis e right-parenthesis With presentation form for vertical right-brace Underscript 0 Endscripts plus one half upper B left-parenthesis e comma e right-parenthesis 3rd Row 1st Column equals 2nd Column pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis plus double-vertical-bar e double-vertical-bar Subscript upper E left-parenthesis upper I right-parenthesis Superscript 2 Baseline period EndLayout

Remark 1.10 Consider the problem given by eq. (1.5) and assume that κ and c are constants. In this case the smoothness of u depends only on the smoothness of f: If Скачать книгу