Our aim is to express, if possible, the discretization error
[e.sub.i] = [u.sub.i] - [U.sub.i] at the ith mesh point in terms of hi.
This coercivity (ellipticity) property together with a corresponding boundedness property and consistency as well as approximation results for the IgA spaces yields the corresponding a priori discretization error
The spatial discretization error
on the h mesh is defined as [[epsilon].sup.h] = [[[psi]].sub.h] - [[psi].sup.h], with [[[psi]].sub.h] being the exact solution to the spatial continuous equation projected onto the discrete domain.
The difference between these two fields sometimes also can be called representation error (RE), since RE can be referred to as forward interpolation error [19, 20] which is subject to the effects of discretization error
and limited resolution .
Figure 3(a) shows the discretization error
on the total field, with h ranging from 0.0352 cm [approximately equal to] [[lambda].sub.1]/224 to 2.25 cm [approximately equal to] [[lambda].sub.1]/3.5.
The errors incurred are then discretization error
and numerical inversion of the Bromwich integral which is O([10.2.sup.-N]) .
To evaluate the discretization error
, we calculated the relative difference (ERR) between the maximal value in the course and the fine time resolution, as follows:
7 the eigenvalues are very close, this slight difference is due to the discretization error
of the geometry in the conventional finite element method while the isogeometric analysis gives an exact geometry.
The mesh procedure in the finite element method (FEM) and boundary element method (BEM) [1-5] presents at least two problems: (i) mesh generation under certain conditions is still arduous, time-consuming, and fraught with pitfalls and (ii) the geometry of the model is approximated by elements and thereby results in additional discretization error
. Meshless (mesh-free) methods have been developed in the past decades to reduce the required effort for mesh generation.
However, in all three cases the resulting digital resonator resonant frequency [[omega].sub.n], due to the discretization error
, differs from the corresponding continuous time domain PR controller resonant frequency [[omega].sub.n].
The discretization error
[E.sub.D] can be made arbitrary small by choosing d large enough .
Karttunen, "Stencils with isotropic discretization error
for differential operators," Numerical Methods for Partial Differential Equations, vol.