Mathematical aspects of finite element I (有限元的数学概念）

 

   Recently, I read a light book <FInite Elemetns Mathematical aspets Volume IV> by, J. T. Oden, G. F. Carey.  This explains why FEM works.

 -- some well-known function spaces :

    * C^m , the linear space consisting of all functions u with partial derivatives D^a u of orders  0 <= |a| <= m continuous 

    * L^p,  the linear space of equivalence classes of measurable functions u for which the \int |u|^p < \inifinite

    * Sobolev space, W^(m,p) of order (m, p) is the linear space of functions in L^p whose distributional derivatives D^a u of all orders |a| such that 0 <= |a| <= m, are in L^p.

     W^(m,p) = { u | D^a u /belong L^p for 0 <= |a| <= m},

      the energy-norm of W usually written as  H^m  = W^(m, 2) 

    * A normal space U is saide to be embedded in a normed space V, if (1) U is a linear subspace of V and (2) the injection operator: U->V is continuous.

     ->  there comes the idea " interpolate " between spaces  H^1, H^0 to define intermediate space H^{\theta}, where  0<= \theta <= 1 

     

 -- fractional spaces ::  relationship between space of functions defined on bounded domains and these boundary spaces 

   * trac operators ::  the normal derivatives of u on boundary spaces    \gamma_j

      \gamma_j  u(x) =  \partial^j u(x) / \partial n^j   

   --> the trace operators \gamma_j can be extended to continuous linear operators mapping H^m on bounded domains onto H^{m - j -1/2)  on boundary spaces .  e.g.  Green function  

  

 --  Quotient Space 

        Q^(m,p) = W^(m,p) / P_(m-1)

        where P_(m-1)  is the space of polynomials in x of degree <= m - 1.

       The elements of Q^(m,p) are cosets [v] of functions such that 

       for u, v \belongs W^(m,p),   u \belongs [v]   ==>  u - v \belongs P_(m-1)

   

 -- variational B. V. P

      let B(. , .) denotes a continuous bilinear form mapping H x G into R and consider a variational boundary-value-problem of finding u \belongs H such that

       B(u, v) = F(v)   for v \belongs G

      note, for each fixed u, B(u, .) defines a continuous line functional A_u on H, and that this functional depends linearly on the choice of u 

     *  approximations 

      the solution u and test function v may belongs to very large class of functions, so the approximation of solutions are based on an elementary ideas: 

      reconstruct the problem so that one need only work with restricted classes of functions

     * Galerkin's method

     Let  H^h (0<h<=1) and G^h (0 < h <=1 ) denote families of closed subspaces of H and G, h is a mesh-base parameter

     so Galerkin approximation of the origin equation involves the problem of find u_h \belongs H^h such that

    B(u_h, v_h) = F(v_h)    for v_h \belongs G^h

    and this approximate problem has a unique solution u*_h if B(. , .) is continue from HxG -> R

    By setting v = v^h in the original equation, and define error as  e =  u - u^h

   we see that the error is orthogonal to G^h   as B(e, v^h) = 0  for v^h \belongs G^h 

    -->  Galerkin approximation is sometimes called the best approximation to u* in H^h

 and the quality of the approximation depends on how well H^h and G^h approximate H and G, actually we will see the approximation error is bounded by a term || u - u^h ||_ h

 -- finite element interpolation theory

     Finite Element is an approximation of a function in terms of its values or values of its derivatives at specified nodal points in the domain of the function, locally. it generally represents the function as a polynomial in much the same spirit as classical Lagrange or Hermite interpolation methods.

     Question 1 :  given a function u belonging to a Sobolev space  W^(m,p), construct a finite element representation of u which approximates u as closely as desired

     Question 2:  estimate the error inherent in the interplant for a given finite element mesh

   

    * Affine families of finite elements

    Question :  are all elements in a family somehow equivalent ?

    if there exist an affine map F, between two elements with the same family,  which mapping points in one element to another, we say these two elements are affine equivalent. 

   --> the master element works

  Question: interpolation errors ?

  First, introduce interpolation operator  P :  W^(m,p)  -->  S^h

  这本书，看的不是很懂。以后有空还要仔细看看。 后两章讲 mixed method 和 hybrid method。