examples/navier-stokes.md

@@ -64,14 +64,15 @@ Implementation
 
 We split the implementation into three parts, 1. the time derivative, 2. the stokes
 operator and 3. any external forces. For addressing the different components of
-the Taylor-Hood basis, we introduce the treepaths `auto _v = Dune::Indices::_0` and
-`auto _p = Dune::Indices::_1` for velocity and pressure, respectively.
+the Taylor-Hood basis, we introduce the treepaths `_v = Dune::Indices::_0` and
+`_p = Dune::Indices::_1` for velocity and pressure, respectively.
 
 ### Problem framework
-We have an instationary problem consistenting of a sequence of stationary equations
+We have an in-stationary problem consisting of a sequence of stationary equations
 for each timestep, thus we combine a `ProblemInstat` with a `Problemstat`.
 
 ```c++
+using namespace AMDiS;
 ProblemStat prob("stokes", grid, basis);
 prob.initialize(INIT_ALL);
 
@@ -99,7 +100,7 @@ prob.addMatrixOperator(opTime, _v, _v);
 
 // <1/tau * u^old, v>
 auto opTimeOld = makeOperator(tag::testvec{},
-  density * invTau * prob.solution(_v));
+  density * invTau * probInstat.oldSolution(_v));
 prob.addVectorOperator(opTimeOld, _v);
 
 for (int i = 0; i < Grid::dimensionworld; ++i) {
@@ -137,3 +138,7 @@ prob.addMatrixOperator(opP, _v, _p);
 auto opDiv = makeOperator(tag::test_divtrialvec{}, 1.0);
 prob.addMatrixOperator(opDiv, _p, _v);
 ```
+where we have used the identity
+```math
+\nabla\mathbf{u}:\nabla\mathbf{v} = \sum_i \nabla u_i\cdot\nabla v_i
+```
\ No newline at end of file