Commit 6d6d8969 authored by Praetorius, Simon's avatar Praetorius, Simon

Some Documentation updated

parent fa24f5f8
......@@ -21,7 +21,7 @@ can be obtained from https://gitlab.dune-project.org and need to be found in a
subdirectory of `DUNE_CONTROL_PATH`. See also https://dune-project.org/doc/installation
for details about the installation of dune modules.
Additionally we require the following libraries to be found:
Additionally, we require/suggest the following libraries to be found:
- [MTL4](https://gitlab.math.tu-dresden.de/spraetor/mtl4) (use this fork to get up-to-date changes)
- [Boost](http://www.boost.org) >= 1.40
- [SuiteSparse](http://faculty.cse.tamu.edu/davis/suitesparse.html) (optional)
......
AMDiS {#mainpage}
=====
The *Adaptive Multi-Dimensional Simulation Toolbox* (AMDiS) is implemented as a
discretization module on top of the [Dune](https://dune-project.org) framework.
Example
-------
An AMDiS program consists of three main incredients:
1. A Problem class that holds all information necessary for assembling a linear
system, see \ref ProblemStat.
2. Operators describing the (bi)linear-form of your PDE, see \ref operators.
3. Adaption-modules for the time- and space adaptive solution of the problem, see \ref Adaption.
**Poisson equation:**
The most simple elliptic PDE is the Poisson equation:
\f{eqnarray*}{
-\Delta u &=& f(x),\quad\mbox{ in }\Omega \\
u &=& g(x),\quad\mbox{ on }\partial\Omega
\f}
where \f$ f(x) \f$ and \f$ g(x) \f$ are parameters describing the volume and
boundary forces of the problem.
The corresponding weak form of the equation reads:
\f[
\langle \nabla v, \nabla u\rangle_\Omega = (f(x)\,v)_\Omega,\quad\forall v\in V_0
\f]
with \f$ u\in V_g \f$.
Thus, we need to define a grid (discretization of \f$ \Omega \f$) and a finite
element space (discretization of \f$ V_0 \f$ and \f$ V_g \f$). This cab be
provided as `Traits` parameter in the ProblemStat class:
~~~~~~~~~~~~~~~{.cpp}
using Grid = Dune::AlbertaGrid<AMDIS_DIM, AMDIS_DOW>;
using Traits = LagrangeBasis<Grid::LeafGridView, 1>;
~~~~~~~~~~~~~~~
where `AlbertaGrid` defines a grid and `Traits` a Lagrange Finite-Element space with
local polynomial degree 1 on the elements.
All AMDiS programs start with initialization of the library, using `AMDiS::init`
~~~~~~~~~~~~~~~{.cpp}
using namespace AMDiS;
int main(int argc, char** argv)
{
AMDiS::init(argc, argv);
ProblemStat<Traits> prob("name");
~~~~~~~~~~~~~~~
The Problem class is initialized with a name, that is used as identifier in the
parameter files. In order to initialize the Finite-Element space, the grid and
all other parts of the problem class, call `initialize(Flag)` where `Flag`
specifies what to initialize. For now, we initialize everything: `INIT_ALL`:
The *Adaptive Multi-Dimensional Simulation Toolbox* (AMDiS) is a Finite-Element
discretization module allowing for fast prototyping of PDEs on adaptively refined
grids. It is implemented on top of the [Dune](https://dune-project.org) framework.
Tutorial
--------
As an example of usage, we want to solve an elliptic PDE, the Poisson equation,
\f$ -\Delta u = f \f$ in \f$ \Omega \f$ with \f$ u = g \f$ on a subset of the boundary
\f$ \Gamma\subset\partial\Omega \f$. For simplicity, we assume \f$ f(x) \equiv -1 \f$
and \f$ g(x) \equiv 0 \f$, the domain \f$ \Omega \f$ a square domain \f$ [0,1]^2 \f$
and \f$ \Gamma \f$ the lower and left edge of the boundary.
~~~~~~~~~~~~~~~{.cpp}
prob.initialize(INIT_ALL);
~~~~~~~~~~~~~~~
Operators specify the (bi-)linear-form and the coefficient function in the term,
see \ref operators for a list of possible types. The bilinear-form in the Poisson
equation consists of a second-order term with coefficient = 1 and the linear-form
includes the function \f$ f(x)=-1 \f$:
~~~~~~~~~~~~~~~{.cpp}
auto opL = makeOperator(tag::gradtest_gradtrial{}, 1.0);
prob.addMatrixOperator(opL, 0, 0);
auto opF = makeOperator(tag::test{}, [](auto const& x) { return -1.0; }, 0);
prob.addVectorOperator(opF, 0);
~~~~~~~~~~~~~~~
Boundary conditions, in the example above a Dirichlet condition, is specified by
defining a predicate for the boundary \f$ \Gamma\subset\partial\Omega \f$ and the
values on the boundary \f$ g(x) = 0 \f$:
~~~~~~~~~~~~~~~{.cpp}
auto predicate = [](auto const& x){ return x[0] < 1.e-8 || x[1] < 1.e-8; };
prob.addDirichletBC(predicate, 0, 0, 0.0);
~~~~~~~~~~~~~~~
The final step is the assembling and solution of the linear system. (Maybe including
grid adaption). This is realized using an \ref AdaptStationary class:
~~~~~~~~~~~~~~~{.cpp}
AdaptInfo adaptInfo("adapt");
AdaptStationary adapt("adapt", prob);
adapt.adapt(); // assemble and solve
~~~~~~~~~~~~~~~
Finally, finish the AMDiS program with `AMDiS::finish()`.
The complete program then reads:
~~~~~~~~~~~~~~~{.cpp}
#include <dune/amdis/AMDiS.hpp>
#include <dune/amdis/AdaptInfo.hpp>
#include <dune/amdis/AdaptStationary.hpp>
#include <dune/amdis/ProblemStat.hpp>
// The namespace all AMDiS classes and functions are defined in
using namespace AMDiS;
// A dune grid type
using Grid = Dune::AlbertaGrid<AMDIS_DIM, AMDIS_DOW>;
// A dune-functions globalBasis wrapped in a struct,
// here representing local polynomial shape functions of degree 1
using Traits = LagrangeBasis<Grid::LeafGridView, 1>;
int main(int argc, char** argv)
int main(int argc, char* argv[])
{
// Initialize linear-algebra backend and read parameters from file
AMDiS::init(argc, argv);
// Create a problem class containing all data for assembling
ProblemStat<Traits> prob("poisson");
// Initialize grid, globalBasis, solution vector and systenmatrix
prob.initialize(INIT_ALL);
// An operator representing the weak laplacian with coefficient = 1.0
auto opL = makeOperator(tag::gradtest_gradtrial{}, 1.0);
prob.addMatrixOperator(opL, 0, 0);
// An rhs-operator representing an analytic function f(x) = -1
auto opF = makeOperator(tag::test{}, [](auto const& x) { return -1.0; }, 0);
prob.addVectorOperator(opF, 0);
// set boundary condition
// Define the boundary Gamma
auto predicate = [](auto const& x){ return x[0] < 1.e-8 || x[1] < 1.e-8; };
// Set a value g(x) = 0 at this part of the boundary
prob.addDirichletBC(predicate, 0, 0, 0.0);
// assemble and solve
// assemble and solve the linear system
AdaptInfo adaptInfo("adapt");
AdaptStationary adapt("adapt", prob);
adapt.adapt();
......@@ -130,4 +59,12 @@ int main(int argc, char** argv)
AMDiS::finalize();
return 0;
}
~~~~~~~~~~~~~~~
\ No newline at end of file
~~~~~~~~~~~~~~~
Notes
-----
An AMDiS program consists of three main incredients:
1. A Problem class that holds all information necessary for assembling a linear
system, see \ref ProblemStat.
2. Operators describing the (bi)linear-form of your PDE, see \ref operators.
3. A parameter file controlling several parts of the solution process, see \ref Initfile.
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