By Peter S. Huyakorn
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Extra resources for Computational Methods in Subsurface Flow
It becomes apparent, when we assemble the elements, that the global basis function associated with each internal node I of the mesh is given by 2 (χ - (X 1+l Χ/-ι)/(χ, - X)/(X I+X */_,) - Xj) for for χ , . ^ χ ^ χ , χ, ^ χ ^ Xj,,. As illustrated in Fig. 7c, Nj takes the shape of a hat. F o r this reason, it is called the " c h a p e a u " basis function. Knowing the expressions for the basis functions TV/ of all nodes in the n e t w o r k , we can determine the distribution of the trial function û over the entire region.
2 Stability Analysis In solving the transient problem using time-stepping m e t h o d s , it is important to e n s u r e that the time-stepping process is computationally stable. B e c a u s e c o m p u t e r s can store only a finite n u m b e r of digits to represent each real number, there exist round-off errors whose cumulative effect is the difference b e t w e e n the c o m p u t e d and the exact solutions of the finite element matrix equation. A computationally stable s c h e m e must be able to control the growth of the cumulative round-off error.
25) w e can readily extract the element equation. 26) where dx e = x In matrix notation, E q . 27) where 'dN, dN dN dx dx dx dx dN dN dN dN dx dx dx dx x x dN£ dx, [CY JX\ 2 x 2 2 fri' It should b e n o t e d that the element matrix obtained from the Galerkin approach is identical to that from the variational approach. Thus, the remaining steps of the finite element solution need no further elaboration. Example 2 As a second e x a m p l e , we consider the problem of twodimensional, horizontal flow in an isotropic reservoir formation (Fig.