Triangular elements in the finite element method

Abstract

For a plane polygonal domain Q and a corresponding (general) triangulation we define classes of functions pmix, v) which are polynomials on each triangle and which are in C^'CQ) and also belong to the Sobolev space ^""^'(n). Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.

Error Analysis of Finite Element Methods With Triangles for Elliptic Boundary Value Problems

“…In [4] the results of Sard [12] are used, and it is through the use of the Sard kernel theorems that we obtain sharp bounds for the interpolation errors in each element, in this case a triangle. The forms of our bounds are similar to those of [22] and [5] which contain unknown constants, but we are able to compute the corresponding constants. The bounds can be used to produce bounds in the Sobolev norm for the finite element solution of the elliptic boundary value problem as in [5].…”

“…The forms of our bounds are similar to those of [22] and [5] which contain unknown constants, but we are able to compute the corresponding constants. The bounds can be used to produce bounds in the Sobolev norm for the finite element solution of the elliptic boundary value problem as in [5].…”

“…Zlamal [5] with triangular elements. In [4] the results of Sard [12] are used, and it is through the use of the Sard kernel theorems that we obtain sharp bounds for the interpolation errors in each element, in this case a triangle.…”

Finite element subspaces with optimal rates of convergence for the stationary Stokes problem

“…Then v h e C(Q) and v h \ r -0 ; so S h e V h . Since v h = v for v a polynomial of degree fc or less, the bound (3.6) follows from well-known finite element error techniques, see [4] and [5], for example.…”

On the numerical solution of plate bending problems by hybrid methods

“…Since Vfô h ) is a closed subspace of S, it will be equipped with the norm (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) \\v\\v = llslloIt will also be convenient for the foliowing to associate to every cp in WÇG h ) a stress tensor M(cp) in K(lS ft ) defined by (1.21) M(q>)y…”