Curved Elements in the Finite Element Method. I

A mixed method for 4th order problems using linear finite elements

“…Let S,, = 5 ft (F A ) be the space of continuous functions which are linear in each triangle of T h with the usual modification for the curved éléments (see Ciarlet-Raviart [2], Zlamal [10]). The space S h is the intersection of S h and W\ 9 The approximation properties of the spaces S h and S h are wellknown; confer Ciarlet-Raviart [2] for example.…”

On the finite element approximation of solutions for radiation problem

“…The uniqueness of solution to (1) and (2) indicates that the solution M £ has the représentation u^d-vl/Q)^!-^)" 1 /. (27) Take E = 0. The existence (1 -T ; ) -1 is seen as in [19] and we omit it In the proof of the following theorem we will see that T k+ie H-• T k , e -• 0.…”

“…[4], [23].) Near the boundary curved éléments are used [26], [27]. If S h dénotes the trial subspace of continuous piecewise linear functions (except over the curved triangles) which vanish on the nodes of the triangulation lying on the boundary of Q(R\ then the error estimate…”

Finite element solution of a nonlinear diffusion problem with a moving boundary

“…We map K(t) on the (time independent) 406 L. CERMÂK, M. ZLÂMAL référence element K with vertices R t = (0, 0), R 2 = (1, 0), R 3 = (0, 1) in the £,, rj -plane. We have (see [7]) :…”

“…We cover Q(0) by triangles completed along F(0) by curved éléments in a manner described in Zlâmal [7]. Let P k = (x h y k ), k = 1,..., d, be the nodes of this triangulation and let Q k = (a fc , (î fc ) be their inverse images in the mapping to 0)…”