Mathematical aspects of finite element I (有限元的数学概念）
Recently, I read a light book
-- 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 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。