nevertheless retains an ambiguity: since only derivatives of the solution
appear in the equations, the solution is only determined up to a constant. For
this reason, we have to pose another condition for the numerical solution,
-which fixes this constant.
+which fixes this constant.
For this, there are various possibilities:
<ol>
functions. Therefore, even though fixing one node is allowed for
discretized functions, it is not for continuous functions, and one can
often see this in a resulting error spike at this point in the numerical
- solution.
-
+ solution.
+
<li> Fixing the mean value over the domain to zero or any other value. This
is allowed on the continuous level, since $H^1(\Omega)\subset L^1(\Omega)$
by Sobolev's inequality, and thus also on the discrete level since we
there only consider subsets of $H^1$.
-
+
<li> Fixing the mean value over the boundary of the domain to zero or any
other value. This is also allowed on the continuous level, since
$H^{1/2}(\partial\Omega)\subset L^1(\partial\Omega)$, again by Sobolev's