Anders Logg  committed ba60012 Merge

Merge branch 'logg/fix-issue-117'

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File demo/documented/neumann-poisson/common.txt

 .. math::
 	- \nabla^{2} u &= f \quad {\rm in} \ \Omega, \\
-    \nabla u \cdot n &= g \quad {\rm on} \ \partial \Omega.
+      \nabla u \cdot n &= g \quad {\rm on} \ \partial \Omega.
 Here, :math:`f` and :math:`g` are input data and :math:`n` denotes the
 outward directed boundary normal. Since only Neumann conditions are
 .. math::
-	\int u \, {\rm d} x = 0
+	\int_{\Omega} u \, {\rm d} x = 0.
 This can be accomplished by introducing the constant :math:`c` as an
 additional unknown (to be sought in :math:`\mathbb{R}`) and the above
+constraint expressed via a Lagrange multiplier.
+We further note that a necessary condition for the existence of a
+solution to the Neumann problem is that the right-hand side :math:`f`
+.. math::
+	\int_{\Omega} f \, {\rm d} x = - \int_{\partial\Omega} g \, {\rm d} s.
+This can be seen by multiplying by :math:`1` and integrating by
+.. math::
+	\int_{\Omega} f \, {\rm d} x = - \int_{\Omega} 1 \cdot \nabla^{2} u \, {\rm d} x
+                                     = - \int_{\partial\Omega} 1 \cdot \partial_n u \, {\rm d} s
+                                       + \int_{\Omega} \nabla 1 \cdot \nabla u \, {\rm d} x
+                                     = - \int_{\partial\Omega} g \, {\rm d} s.
+This condition is not satisfied by the specific right-hand side chosen
+for this test problem, which means that the partial differential
+equation is not well-posed. However, the variational problem expressed
+below is well-posed as the Lagrange multiplier introduced to satisfy
+the condition :math:`\int_{\Omega} u \, {\rm d} x = 0` *effectively
+redefines the right-hand side such that it safisfies the necessary
+condition* :math:`\int_{\Omega} f \, {\rm d} x = -
+\int_{\partial\Omega} g \, {\rm d} s`.
 Our variational form reads: Find :math:`(u, c) \in V \times R` such
 						+ \int_{\Omega} cv \, {\rm d} x
 						+ \int_{\Omega} ud \, {\rm d} x, \\
 	L(v)    &= \int_{\Omega} f v \, {\rm d} x
-    	     	+ \int_{\Gamma_{N}} g v \, {\rm d} s.
+    	     	+ \int_{\partial\Omega} g v \, {\rm d} s.
 :math:`V` is a suitable function space containing :math:`u` and
 :math:`v`, and :math:`R` is the space of real numbers.
 The expression :math:`a(\cdot, \cdot)` is the bilinear form and
 :math:`L(\cdot)` is the linear form.
+Note that the above variational problem may alternatively be expressed
+in terms of the modified (and consistent) right-hand side
+:math:`\tilde{f} = f - c`.
 In this demo we shall consider the following definitions of the domain
 and input functions: