Finite Element Analysis. Barna Szabó

Finite Element Analysis

alt="f element-of upper C Superscript k Baseline left-parenthesis upper I right-parenthesis"/> then u element-of upper C Superscript k plus 2 Baseline left-parenthesis upper I right-parenthesis

u element-of upper C Superscript k plus 2 Baseline left-parenthesis upper I right-parenthesis

for any

. Similarly, if f element-of upper H Superscript k Baseline left-parenthesis upper I right-parenthesis

then

u element-of upper H Superscript k plus 2 Baseline left-parenthesis upper I right-parenthesis

for any

. This is known as the shift theorem. More generally, the smoothness of u depends on the smoothness of κ, c and F. For a precise statement and proof of the shift theorem we refer to [21].

Remark 1.11 An introductory discussion on how a priori estimates are obtained under the assumption that the second derivative of the exact solution is bounded can be found in Appendix B.

1.5.3 A posteriori estimation of error

The goal of finite element computations is to estimate certain quantities of interest (QoIs) such as, for example, the maximum and minimum values of u or u prime on upper I equals left-parenthesis 0 comma script l right-parenthesis . Since finite element solutions are approximations to an exact solution, it is not sufficient to report the value of a QoI computed from the finite element solution. It is also necessary to provide an estimate of the relative error in the QoI, or present evidence that the relative error in the QoI is not greater than an acceptable value.

In this section we will use the a priori estimates described in Section 1.5.2 to obtain a posteriori estimates of error in energy norm. It is possible to obtain very accurate estimates for a large class of problems which includes most problems of practical interest.

Error estimation based on extrapolation

For most practical problems the estimate (1.92) is sufficiently sharp so that the less than or equal sign (≤) can be replaced by the approximately equal sign (≈) and this a priori estimate can be used in an a posteriori fashion.

The computed values of the potential energy corresponding to a sequence of finite element spaces upper S 1 subset-of upper S 2 subset-of midline-horizontal-ellipsis upper S Subscript n can be used for estimating the error in energy norm by extrapolation. Sequences of finite element spaces that have this property are called hierarchic sequences. By Theorem 1.5 and eq. (1.92) we have:

(1.95) pi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis minus pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis almost-equals StartFraction script upper C squared Over upper N Superscript 2 beta Baseline EndFraction

where script upper C equals Overscript def Endscripts upper C double-vertical-bar u Subscript upper E upper X Baseline double-vertical-bar Subscript upper E left-parenthesis upper I right-parenthesis . There are three unknowns: pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis , and β. Assume that we have a sequence of solutions corresponding to the hierarchic sequence of finite element spaces upper S Subscript i minus 2 Baseline subset-of upper S Subscript i minus 1 Baseline subset-of upper S Subscript i . Let us denote the corresponding computed potential energy values by pi Subscript i minus 2 , pi Subscript i minus 1 , πi and the degrees of freedom by upper N Subscript i minus 2 , upper N Subscript i minus 1 , Ni. We will denote the estimate for pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis by π∞. With this notation we have:

(1.96) pi Subscript i Baseline minus pi Subscript infinity Baseline almost-equals StartFraction script upper C squared Over upper N Subscript p Superscript 2 beta Baseline EndFraction

(1.97) pi Subscript i minus 1 Baseline minus pi Subscript infinity Baseline almost-equals StartFraction script upper C squared Over upper N Subscript i minus 1 Superscript 2 beta Baseline EndFraction dot

On dividing eq. (1.96) with eq. (1.97) and taking the logarithm we get

(1.98) log StartFraction pi Subscript i Baseline minus pi Subscript infinity Baseline Over pi Subscript i minus 1 Baseline minus pi Subscript infinity Baseline EndFraction almost-equals 2 beta log StartFraction upper N Subscript i minus 1 Baseline Over upper N Subscript i Baseline EndFraction

and, repeating with i minus 1 substituted for i, it is possible to eliminate 2 beta to obtain:

(1.99) StartFraction pi Subscript i Baseline minus pi Subscript infinity Baseline Over pi Subscript i minus 1 Baseline minus pi Subscript infinity Baseline EndFraction almost-equals left-parenthesis StartFraction pi Subscript i minus 1 Baseline minus pi Subscript infinity Baseline Over pi Subscript i minus 2 Baseline minus pi Subscript infinity Baseline EndFraction right-parenthesis Superscript upper Q

where

upper Q equals log StartFraction upper N Subscript i minus 1 Baseline Over upper N Subscript i Baseline EndFraction left-parenthesis log StartFraction upper N Subscript i minus 2 Baseline Over upper N Subscript i minus 1 Baseline EndFraction right-parenthesis Superscript negative 1 dot

Equation

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