By E. Poisson
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Extra resources for A Relativist's Toolkit - The Math of Black Hole Mechanics
12. A suitable mesh generator is used to subdivide the computational domain fi into element domains Oe. In two dimensions, meshes generally consist of triangular and/or quadrilateral elements. An interesting feature of the finite element method is that it handles naturally unstructured meshes, which can concentrate the elements in regions such as internal or boundary layers where sharp solution gradients are expected. The term "unstructured mesh" means that the number of elements meeting at a node (the element vertices) may vary from node to node.
Then they are introduced in (2. 17) to yield the following discrete equation at an interior node A, A = 2,... : n e q + 1: (2,8) A linear element in ID is defined by two nodes, nen = 2, locally denoted as 1 and 2. The shape functions of a linear element are given by where £ is the normalized coordinate, — 1 < £ < + ! As usual, at any interior point of the element one has w(£) = JVi(Oui +7V 2 (Ou 2 ,andx(0 = JV*i(0xi +^V 2 (0^2Note that for a uniform mesh of size h dx= -(x2 - x i ) d ^ = -df, and thus 8Nb dNb d£ 2dNb ~x~ ox ~ ~*TW o£ ox ~ T~*T h o£ for 6 =1,2.
Here these spaces are defined in the context of the standard Galerkin formulation. The first collection of functions, denoted by V, is composed of test functions and consists of all functions which are square integrable, have square integrable first 22 INTRODUCTION AND PRELIMINARIES derivatives over the computational domain Q, and vanish on the Dirichlet portion, YD, of the boundary. It is defined as follows: This is as previously noted a Sobolev space and its inner product and norm coincide with those of 'H* (SI).