From: schrage Date: Thu, 29 Apr 1999 13:55:08 +0000 (+0000) Subject: Added a section on constraints. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f08a141a5fb6ac252f91c893a0d0a90baa435e3e;p=dealii-svn.git Added a section on constraints. git-svn-id: https://svn.dealii.org/trunk@1221 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/doc/tutorial/chapter-1.elements/hanging_nodes.html b/deal.II/doc/tutorial/chapter-1.elements/hanging_nodes.html index 0712c7d89d..2f3108b1a5 100644 --- a/deal.II/doc/tutorial/chapter-1.elements/hanging_nodes.html +++ b/deal.II/doc/tutorial/chapter-1.elements/hanging_nodes.html @@ -49,6 +49,66 @@ This is further discussed in the chapters on alt="Hanging Nodes"> + +

Constraints on Hanging Nodes

+ +

Where do the constraints come from ?

+

+In order to get constraints for your hanging nodes you have to assume +some function on the boundaries of your cells, as mentioned above. +Take a linear function on the cell boundary between vertices 1 and 2 +with function values x1 and x2. If you get +a hanging node on this boundary during refinement it will be situated +in the middle and therefore its value - the value of the function on +the boundary at this point - will be x3 = 1/2 (x1+x2). +

+ +

Mathematical implications, or
+why is it a constraint matrix ?

+ +

+Assume a system Au=f with u being the solution +vector with all the degrees of freedom of the finite elements, in particular +base functions associated with hanging nodes, which are not true degrees of +freedom of the system of equations. +

+

+The constraint for hanging nodes now is that we must be able to calculate +their value from the values of the surrounding nodes by interpolation. +Let y be a vector with entries corresponding to true degrees of +freedom only. We must be able to write u as
+u=Cy
+where C, the constraint matrix is a square matrix containing the +interpolation. +

+

+Now we have to solve +

+
+
+  Ax = f
+
+with the constraint, that there is a y satisfying
+
+  u = Cy
+
+From this we get
+
+  CTACy = CTf
+
+or, taking
+
+  ~A := CTAC, b := CTf 
+  ~A y = b.
+
+
+ +

+It is not possible to generate ~A directly, but +w can generate A and condense it to ~A +using the constraints, solve the system and obtain u by +u=Cf. +