:feelpp: Feel++
= feelpp_qs_laplacian(1)
Christophe Prud'homme <https://github.com/prudhomm[@prudhomm]>; Vincent Chabannes <https://github.com/vincentchabannes[@vincentchabannes]>
:manmanual: feelpp_qs_laplacian
:man-linkstyle: pass:[blue R < >]
:stem: latexmath

== NAME

{manmanual} - solving Poisson's equation {feelpp}


== SYNOPSIS

{manmanual} --config-file 

== DESCRIPTION

{manmanual} is a simple application which can solve Poisson's equations with Dirichlet, Neumann or Robin boundary condition or a mix of them.

The coefficient of the equations can be automatically computed to get a manufactured solution in order to verify the code with stem:[L^2] and stem:[H^1].

=== Options

{manmanual} requires some options.

.Table of command-line {manmanual} options
|===                                                                                                                                                                              
| Name | Description | Default value

| config-file | .cfg file to configure the application |  
| checker.check | check the norms with respect to a manufactured solution | true
| checker.compute-pde-coefficients | compute pde coefficients from manufactured solution | true
| checker.solution | manufactured solution |
| checker.gradient | gradient of the manufactured solution |
| f | right hand side expression |
| un | expression on Neumann boundary | 
| g | expression on Dirichlet boundary |
| r_2 | right hand side expression on Robin/Fourier boundary |
|===  

== EXAMPLES

We are now using the `feelpp/feelpp-toolboxes:latest` docker image  as described in link:http://book.feelpp.org/user-manual/#getting_started[] to demonstrate {manmanual} usage.               

NOTE: `feelpp/feelpp` docker image works as well.

[source,shell]
.Starting the docker `feelpp/feelpp-toolboxes:latest`
----
docker run --rm -ti -v $HOME/feel:/feel feelpp/feelpp-toolboxes
----

=== Solving Poisson's equation in a triangle

Let's consider Poisson's equation in a triangle with each side associated respectively to Dirichlet, Neumann and Robin condition.

==== Solve using manufactured solution

The application can be run to solve the equation and verify the solution using a manufactured solution using `--checker.check=true` and `--checker.solution=<expr>` to provide the solution.

The coefficients associated to the manufactured solution can be computed automatically by a python script (`--checker.compute-pde-coefficients=true`) or provided by the user(`--checker.compute-pde-coefficients=false`)

.Solving Poisson's equation using manufacture solution and let {feelpp} compute the coefficients of the PDE
[source,shell]
----
 ./feelpp_qs_laplacian_2d --config-file laplacian/triangle/triangle.cfg --checker.solution="x" --checker.compute-pde-coefficients=true 
----

.Solving Poisson's equation using manufacture solution and provide the coefficients of the PDE
[source,shell]
----
 ./feelpp_qs_laplacian_2d --config-file laplacian/triangle/triangle.cfg --checker.solution="x" --checker.compute-pde-coefficients=false --un=-nx:x --g=x:x --r_2=-nx-x:nx:x --f=1
----

==== Solving Poisson's equation

Poisson's equation can be solved by simply giving the material and boundary condition expressions using `--checker.check=false`.
In that case, the solution may in general not be known and no verification is done.

.Solving Poisson's equation with the coefficients provided
[source,shell]
----
 ./feelpp_qs_laplacian_2d --config-file laplacian/triangle/triangle.cfg --checker.compute-pde-coefficients=false --ckecker.check=false --un=0 --g=0 --r_2=0 --f=1
----

== SEE ALSO

*{feelpp} Book:* http://book.feelpp.org
*{feelpp} mathemmatics:* http://docs.feelpp.org/math/current/fem/laplacian/

== COPYING

Copyright \(C) 2017-2020 {feelpp} Consortium. +
Free use of this software is granted under the terms of the GPLv3 License.