On the finite element method

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].…”

“…Zlamal [22] using first quadratic and then cubic trial functions shows that Bramble and Zlamal for a typical triangle T show that, if P 4m+1 (x,y) interpolates u(x,y), and it is assumed that…”

“…We first mention briefly the techniques of Birkhoff, Schultz and Varga [4] for obtaining bounds when using rectangular elements, and those of Zlamal [22] and Bramble and…”

On a Bisection Algorithm That Produces Conforming Locally Refined Simplicial Meshes

“…Several variants of the algorithm suitable for standard FEMs were also proposed, analysed and numerically tested in [1,2,3,8,10,11] (see also references therein). It has been commonly noticed that inspite of a general simplicity of this type of bisection algorithms, it turns to be hard to provide mesh conformity and simultaneously to prove relevant mesh regularity results [4,20], especially in the case of local mesh refinements and in higher dimensions.…”

Higher order approximations to the boundary conditions for the finite element method

“…[3], [10], [14], [16] showing that ||0||, ,2 r < CA||0||j n for all 0 G F" and applying Theorem 2. The inequality above can be deduced by the same method discussed in Section 5.1.…”