From: bangerth Date: Thu, 7 Sep 2006 22:18:17 +0000 (+0000) Subject: Fix up formulas. Reference step-23 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0ba9272d3449d9d1724048f2fe7807bc013d86c9;p=dealii-svn.git Fix up formulas. Reference step-23 git-svn-id: https://svn.dealii.org/trunk@13853 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/deal.II/include/numerics/vectors.h b/deal.II/deal.II/include/numerics/vectors.h index b2876ed662..705241e595 100644 --- a/deal.II/deal.II/include/numerics/vectors.h +++ b/deal.II/deal.II/include/numerics/vectors.h @@ -91,19 +91,19 @@ class ConstraintMatrix; * given function may be, taking into account that a virtual function has * to be called. * - *
  • Projection: compute the L2-projection of the given - * function onto the finite element space, i.e. if f is the function - * to be projected, compute fh in L2 such - * that (fh,vh)=(f,vh) for all discrete - * test functions vh. This is done through the solution of the - * linear system of equations $M v = f$ where $M$ is the mass matrix - * $m_{ij} = \int_\Omega \phi_i(x) \phi_j(x) dx$ and - * $f_i = \int_\Omega f(x) \phi_i(x) dx$. The solution vector $v$ then is - * the nodal representation of the projection fh. + *
  • Projection: compute the $L^2$-projection of the given function + * onto the finite element space, i.e. if $f$ is the function to be + * projected, compute $f_h\in V_h$ such that $(f_h,v_h)=(f,v_h)$ for + * all discrete test functions $v_h$. This is done through the + * solution of the linear system of equations $M v = f$ where $M$ is + * the mass matrix $m_{ij} = \int_\Omega \phi_i(x) \phi_j(x) dx$ and + * $f_i = \int_\Omega f(x) \phi_i(x) dx$. The solution vector $v$ then + * is the nodal representation of the projection $f_h$. The project() + * functions are used in the @ref step_23 "step-23" tutorial program. * * In order to get proper results, it be may necessary to treat * boundary conditions right. Below are listed some cases where this - * may be needed. If needed, this is done by L2-projection of + * may be needed. If needed, this is done by $L^2$-projection of * the trace of the given function onto the finite element space * restricted to the boundary of the domain, then taking this * information and using it to eliminate the boundary nodes from the @@ -133,10 +133,10 @@ class ConstraintMatrix; * * Obviously, the results of the two schemes for projection are * different. Usually, when projecting to the boundary first, the - * L2-norm of the difference between original + * $L^2$-norm of the difference between original * function and projection over the whole domain will be larger * (factors of five have been observed) while the - * L2-norm of the error integrated over the + * $L^2$-norm of the error integrated over the * boundary should of course be less. The reverse should also hold * if no projection to the boundary is performed. * @@ -217,7 +217,7 @@ class ConstraintMatrix; * The @p project_boundary_values function acts similar to the * @p interpolate_boundary_values function, apart from the fact that it does * not get the nodal values of boundary nodes by interpolation but rather - * through the L2-projection of the trace of the function to the boundary. + * through the $L^2$-projection of the trace of the function to the boundary. * * The projection takes place on all boundary parts with boundary * indicators listed in the map (FunctioMap@p ::FunctionMap) @@ -275,14 +275,14 @@ class ConstraintMatrix; * use of the wrong quadrature formula may show a significantly wrong result * and care should be taken to chose the right formula. * - * The H1 seminorm is the L2 + * The $H^1$ seminorm is the $L^2$ * norm of the gradient of the difference. The square of the full - * H1 norm is the sum of the square of seminorm - * and the square of the L2 norm. + * $H^1$ norm is the sum of the square of seminorm + * and the square of the $L^2$ norm. * * To get the global L1 error, you have to sum up the * entries in @p difference, e.g. using - * Vector::l1_norm function. For the global L2 + * Vector::l1_norm function. For the global $L^2$ * difference, you have to sum up the squares of the entries and * take the root of the sum, e.g. using * Vector::l2_norm. These two operations @@ -294,8 +294,8 @@ class ConstraintMatrix; * To get the $L_\infty$ norm, take the maximum of the vector elements, e.g. * using the Vector::linfty_norm function. * - * For the global H1 norm and seminorm, the same rule applies as for the - * L2 norm: compute the $l_2$ norm of the cell error vector. + * For the global $H^1$ norm and seminorm, the same rule applies as for the + * $L^2$ norm: compute the $l_2$ norm of the cell error vector. * * * All functions use the finite element given to the DoFHandler object the last