Finite Element Analysis. Barna Szabó

Finite Element Analysis

(1.99) can be solved for π∞ to obtain an estimate for the exact value of the potential energy.

The relative error in energy norm corresponding to the ith finite element solution in the sequence is estimated from

(1.100) e Subscript i Baseline almost-equals left-parenthesis StartFraction pi Subscript i Baseline minus pi Subscript infinity Baseline Over StartAbsoluteValue pi Subscript infinity Baseline EndAbsoluteValue EndFraction right-parenthesis Superscript 1 slash 2 Baseline period

Usually the percent relative error is reported. This estimator has been tested against the known exact solution of many problems of various smoothness. The results have shown that it works well for a wide range of problems, including most problems of practical interest; however, it cannot be guaranteed to work well for all conceivable problems. For example, this method would fail if the exact solution would happen to be energy‐orthogonal to all basis functions associated with (say) odd values of i.

Remark 1.12 From equation (1.92) we get

(1.101) log left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E Baseline almost-equals log upper C minus beta log upper N period

On plotting left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E vs. N on log‐log scale a straight line with the slope negative beta will be seen for sufficiently large N. The estimated value of β, corresponding to the ith solution in the sequence, is denoted by βi. It is computed from eq. (1.98):

(1.102) beta Subscript i Baseline equals one half StartFraction log left-parenthesis pi Subscript i Baseline minus pi right-parenthesis minus log left-parenthesis pi Subscript i minus 1 Baseline minus pi right-parenthesis Over log upper N Subscript i minus 1 Baseline minus log upper N Subscript i Baseline EndFraction dot

Examples

The properties of the finite element solution with reference to a family of model problems is discussed in the following. The problems are stated as follows: Find u Subscript upper F upper E Baseline element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis such that

(1.103) integral Subscript 0 Superscript script l Baseline left-parenthesis kappa u prime Subscript upper F upper E Baseline v Superscript prime Baseline plus c u Subscript upper F upper E Baseline v right-parenthesis d x equals upper F left-parenthesis v right-parenthesis for all v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis

where κ and c are constants and upper F left-parenthesis v right-parenthesis is defined such that the exact solution is:

(1.104) u Subscript upper E upper X Baseline equals x Superscript alpha Baseline left-parenthesis script l minus x right-parenthesis comma on upper I equals left-parenthesis 0 comma script l right-parenthesis comma alpha greater-than 1 slash 2 period

As explained in Section 1.5.1, when α is not an integer, the case considered in the following, then this solution lies in the space upper H Superscript alpha plus 1 slash 2 minus epsilon Baseline left-parenthesis upper I right-parenthesis . Therefore the asymptotic rate of h‐convergence on uniform meshes, predicted by eq. (1.92), is beta equals alpha minus 1 slash 2 and the asymptotic rate of p‐convergence on a fixed mesh is beta equals 2 alpha minus 1 .

We selected this problem because it is representative of the singular part of the exact solutions of two‐and three‐dimensional elliptic boundary value problems.

Referring to Theorem 1.3, we have upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma v right-parenthesis equals 0 for all v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis therefore upper F left-parenthesis v right-parenthesis equals upper B left-parenthesis u Subscript upper E upper X Baseline comma v right-parenthesis . Consequently for the kth element the load vector in the local numbering convention is:

(1.105)

where by definition phi Subscript i Baseline left-parenthesis upper Q Subscript k Baseline left-parenthesis xi right-parenthesis right-parenthesis equals upper N Subscript i Baseline left-parenthesis xi right-parenthesis .

When 1 slash 2 less-than alpha less-than 1 then the first derivative of u Subscript upper E upper X is infinity in the point x equals 0 . To avoid having u prime Subscript upper E upper X in the integrand, the first term in eq. (1.105) is integrated by parts:

integral Subscript x Subscript k Baseline Superscript x Subscript k plus 1 Baseline Baseline kappa u prime Subscript upper E upper X Baseline phi Subscript i Superscript prime Baseline d x equals left-parenthesis kappa u Subscript upper E upper X Baseline phi prime Subscript i right-parenthesis Subscript x Sub Subscript k Subscript Superscript x Super Subscript k plus 1 Superscript Baseline minus integral Subscript x Subscript k Baseline Superscript x Subscript <hr><noindex><a href=

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