# Problem

## Summary

The `ProblemStat`

and `ProblemInstat`

are classes collecting everything necessary to assemble stationary or instationary
PDEs, respectively.

A `ProblemStat`

contains the data structures for the linear
system, i.e. system matrix, solution and right-hand side vector, it contains the
grid and global basis, a linear solver, marker and error estimator, and much more,
see below.

A `ProblemInstat`

is responsible for storing data needed for time discretization,
like an old-solution vector, references to time, timestep width and the stationary
problem to solve in each timestep.

These two classes build the basis for most AMDiS projects.

### Problem classes

Class | Descriptions |
---|---|

`ProblemStat` |
Standard container for stationary problems (class template) |

`ProblemInstat` |
Standard container for instationary problems (class template) |

### Interfaces

Interface | Descriptions |
---|---|

`StandardProblemIteration` |
A master problem for a single non coupled problem. (class) |

`ProblemStatBase` |
Base class for stationary problems. (class) |

`ProblemInstatBase` |
Base class for instationary problems. (class) |

`ProblemTimeInterface` |
Interface for time dependent problems. (abstract class) |

`ProblemIterationInterface` |
Interface for iterations of stationary problems (abstract class) |

## class `ProblemStat`

Defined in header `<amdis/ProblemStat.hpp>`

```
template <class Traits>
class ProblemStat
: public ProblemStatBase
, public StandardProblemIteration
```

`ProblemStat`

is a structure storing data structures for stationary problems.

The template parameter `Traits`

defines the global basis and with this the GridView
and Grid. It is required to have at least the member types `GlobalBasis`

and `CoefficientType`

and may provide additionally a static `create(GridView)`

function to contruct the global basis
from a GridView object. See `DefaultProblemTraits`

and `DefaultBasisCreate`

for
a detailed reference.

### Member Types

Member Type | Definition |
---|---|

`GlobalBasis` |
`typename Traits::GlobalBasis` |

`GridView` |
`typename GlobalBasis::GridView` |

`Grid` |
`typename GridView::Grid` |

`WorldMatrix` |
`FieldMatrix<K, dow, dow>` |

`WorldVector` |
`typename Element::Geometry::GlobalCoordinate` |

`SystemMatrix` |
`DOFMatrix<GlobalBasis, GlobalBasis, C>` |

`SystemVector` |
`DOFVector<GlobalBasis, C>` |

Here, `K`

is the coordinate type, i.e. `K = typename Grid::ctype`

, the constant `dow`

is the
world dimension, i.e. `dow = Grid::dimensionworld`

, and the type `C`

is the coefficients type,
given in the `Traits`

as `C = typename Traits::CoefficientType`

.

### Member functions

Function | Descriptions |
---|---|

(constructor) |
Construct the problem |

`initialize` |
Initializes all data-structures in the problem |

`addMatrixOperator` |
Adds an operator to the system matrix |

`addVectorOperator` |
Adds an operator to the rhs vector |

`addDirichletBC` |
Adds a Dirichlet boundary condition to the system |

`addPeriodicBC` |
Adds a periodic boundary condition to the system |

`writeFiles` |
Writes output files. |

#### Getters and Setters

Function | Descriptions |
---|---|

`name` |
The name of the problem |

`grid, setGrid` |
Get and set the stored grid |

`gridView` |
Get the GridView stored in the global basis |

`boundaryManager` |
Return the boundary manager to identify boundary segments |

`globalBasis` |
Return the stored global basis |

`solver, setSolver` |
Get and set the linear solver |

`systemMatrix, solutionVector, rhsVector` |
Return the matrix and vector of the linear system |

`solution` |
Return a grid function of the solution |

#### Implementation of the `ProblemStatBase`

interface

Function | Descriptions |
---|---|

`buildAfterAdapt, assemble` |
Assembles the linear system after grid operations. |

`markElements` |
Marks mesh elements for refinement and coarsening. |

`adaptGrid` |
Refinement/coarsening of the grid. |

`globalCoarsen` |
Uniform global grid coarsening. |

`globalRefine` |
Uniform global refinement. |

`solve` |
Solves the assembled system. |

`estimate` |
A-posteriori error estimation. |

#### Implementation of the `ProblemIterationInterface`

interface

Function | Descriptions |
---|---|

`oneIteration` |
A single build-solve-adapt step |

## function `ProblemStat::ProblemStat`

```
explicit ProblemStat(std::string const& name) // (1)
ProblemStat(std::string const& name, Grid& grid) // (2)
ProblemStat(std::string const& name, Grid& grid, GlobalBasis& globalBasis) // (3)
```

(1) Construct the problem from the grid type and basis type given in the `Traits`

template parameter. Thereby it is assumed, that the `Traits`

type provides a
static `create(GridView)`

function to create the basis. The grid is created by
inspecting the parameter file and using the `MeshCreator`

class.

(2) Construct the problem with given grid. Store the grid reference into a non-destroying
shared_ptr and create the basis from the `Traits`

class, see (1).

(3) Construct the problem with given grid and basis, by storing both references into non-destroying shared_ptr.

#### Arguments

`std::string name`

- The name of the problem that is used to identify parameters in the initfile.
`Grid grid`

- A grid implementing the
`Dune::Grid`

interface. `GlobalBasis globalBasis`

- A basis implementing the
`Dune::Functions::GlobalBasis`

interface.

#### Example

```
using Grid = Dune::YaspGrid<2>;
Grid grid({1.0, 1.0}, {2, 2});
using namespace Dune::Functions::BasisFactory;
auto basis = makeBasis(grid.leafGridView(), lagrange<2>());
// use predefined traits type `LagrangeBasis`
ProblemStat<LagrangeBasis<Grid, 2>> prob1("prob");
// use predefined traits type `LagrangeBasis` but provide a grid directly
// The grid type must match the one defined in the Traits class.
ProblemStat<LagrangeBasis<Grid, 2>> prob2("prob", grid);
// use predefined traits type `LagrangeBasis` but provide a grid and basis directly
// The grid and basis type must match those defined in the Traits class.
ProblemStat<LagrangeBasis<Grid, 2>> prob3("prob", grid, basis);
// using c++17 class template argument deduction
ProblemStat prob4("prob", grid, basis);
```

## function `ProblemStat::initialize`

```
void initialize(Flag initFlag, ProblemStat* adoptProblem = nullptr, Flag adoptFlag = INIT_NOTHING)
```

Initialization of the problem and its data members.

Constructs the grid and basis (if not yet provided otherwise) and creates a linear system data-structures, i.e. system matrix, solution vector and right-hand side vector, based on the size of the basis.

#### Arguments

`Flag initFlat`

- A flag indicating what to initialize, use
`INIT_ALL`

to initialize everything. See below for possible values. `ProblemStat* adoptProblem`

- A problem from which to adopt data not initialized in this problem.
`Flag adoptFlag`

- A flag indicating what to adopt from the
`adoptProblem`

.

#### Init Flags

Flag | Descriptions |
---|---|

`INIT_FE_SPACE` |
Initialize the global basis |

`INIT_MESH` |
Initialize the grid |

`CREATE_MESH` |
Create the grid |

`INIT_SYSTEM` |
Initialize system matrix, solution vector and right-hand side vector |

`INIT_SOLVER` |
Initialize a linear solver |

`INIT_ESTIMATOR` |
Initialize an error estimator |

`INIT_MARKER` |
Initialize a grid marker |

`INIT_FILEWRITER` |
Initialize a file writer |

`INIT_NOTHING` |
Do not initialize anything |

`INIT_ALL` |
Initialize everything |

## function `ProblemStat::addMatrixOperator`

```
// (1)
template <class Operator, class RowTreePath = RootTreePath, class ColTreePath = RootTreePath>
void addMatrixOperator(Operator const& op, RowTreePath row = {}, ColTreePath col = {})
// (2)
template <class Operator, class RowTreePath = RootTreePath, class ColTreePath = RootTreePath>
void addMatrixOperator(BoundaryType b, Operator const& op, RowTreePath row = {}, ColTreePath col = {})
```

(1) Add a local operator, evaluated on the elements, to the system matrix.

(2) Add a local operator, evaluated on boundary intersections, to the system matrix.

#### Arguments

`Operator op`

- A (pre-) local operator to assemble on the local context, see
`LocalOperator`

and`GridFunctionOperator`

. `RowTreePath row`

- TreePath identifying the sub-basis in the global basis tree corresponding
to the row basis, see
`makeTreePath()`

`ColTreePath col`

- TreePath identifying the sub-basis in the global basis tree corresponding
to the column basis, see
`makeTreePath()`

`BoundaryType b`

- Boundary indentifier/number to set on which part of the boundary to assemble
this operator. Can be constructed from an integer, see
`BoundaryType`

.

Note

If no `row`

or `col`

tree-path is given, the root tree-path is assumed, identifying
the root node in the basis tree.

The tree-paths can be constructed from integers or integral-constants, or by using
the `makeTreePath()`

function. It is then transformed into
a `Dune::TypeTree::HybridTreePath<...>`

type.

#### Examples

```
auto op1 = makeOperator(tag::gradtest_gradtrial{}, alpha);
prob.addMatrixOperator(op1, 0, 0);
auto op2 = makeOperator(tag::test_trial{}, beta);
prob.addMatrixOperator(BoundaryType{1}, op2, 0, 0);
```

#### See Also

`DOFMatrix::addOperator`

`addVectorOperator`

## function `ProblemStat::addVectorOperator`

```
// (1)
template <class Operator, class TreePath = RootTreePath>
void addVectorOperator(Operator const& op, TreePath path = {})
// (2)
template <class Operator, class TreePath = RootTreePath>
void addVectorOperator(BoundaryType b, Operator const& op, TreePath path = {})
```

(1) Add a local operator, evaluated on the elements, to the right-hand side vector of the linear system.

(2) Add a local operator, evaluated on boundary intersections, to the right-hand side vector of the linear system

#### Arguments

`Operator op`

- A (pre-) local operator to assemble on the local context, see
`LocalOperator`

and`GridFunctionOperator`

. `TreePath path`

- TreePath identifying the sub-basis in the global basis tree
corresponding to the row basis, see
`makeTreePath()`

`BoundaryType b`

- Boundary indentifier/number to set on which part of the boundary to assemble
this operator. Can be constructed from an integer, see
`BoundaryType`

.

Note

If no `path`

tree-path is given, the root tree-path is assumed, identifying
the root node in the basis tree.

The tree-path can be constructed from integers or integral-constants, or by using
the `makeTreePath()`

function. It is then transformed into
a `Dune::TypeTree::HybridTreePath<...>`

type.

#### Examples

```
auto op = makeOperator(tag::test{}, [g](auto const& x) { return g(x); });
prob.addVectorOperator(BoundaryType{1}, op, _0);
```

#### See Also

`DOFVector::addOperator`

`addMatrixOperator`

## function `ProblemStat::addDirichletBC`

```
// (1)
template <class Predicate, class RowTreePath, class ColTreePath, class Values>
void addDirichletBC(Predicate const& predicate,
RowTreePath row, ColTreePath col,
Values const& values)
// (2)
template <class RowTreePath, class ColTreePath, class Values>
void addDirichletBC(BoundaryType id,
RowTreePath row, ColTreePath col,
Values const& values)
```

Add a Dirichlet boundary condition to the linear system.

(1) The boundary part is identified using a boolean predicate function that is evaluated on the barycenters of the boundary intersections

(2) The boundary part is identified using a boundary identifier/number that was
set in the `BoundaryManager`

before.

#### Arguments

`Predicate predicate`

- A functor
`bool(WorldVector)`

returning true, if the boundary condition should be assembled for the DOFs on the boundary intersection. `RowTreePath row`

- TreePath identifying the sub-basis in the global basis tree corresponding
to the row basis, see
`makeTreePath()`

`ColTreePath col`

- TreePath identifying the sub-basis in the global basis tree corresponding
to the column basis, see
`makeTreePath()`

`Values values`

- A
`GridFunction`

evaluating to the Dirichlet values that should be set for the identified DOFs. `BoundaryType id`

- Boundary indentifier/number to set on which part of the boundary to assemble
this boundary condition. Can be constructed from an integer, see
`BoundaryType`

.

#### Examples

```
// add a Dirichlet BC at the left boundary of the domain [0,1]^2
prob.addDirichletBC([](auto const& x) { return x[0] < 1.e-8; },
0, 0,
[](auto const& x) { return 0.0; });
// same as above but using the boundary id
prob.boundaryManager()->setBoxBoundary({1,2,2,2});
prob.addDirichletBC(BoundaryType{1}, 0, 0, 0.0);
```

## class `ProblemInstat`

Defined in header `<amdis/ProblemInstat.hpp>`

```
template <class Traits>
class ProblemInstat
: public ProblemInstatBase
```

Standard implementation of ProblemTimeInterface for a time dependent problems.

### Member Functions

Function | Descriptions |
---|---|

(constructor) |
Constructs a `ProblemInstat` from a stationary problem. |

`initialize` |
Initialization of the instationary problem. |

`initTimestep` |
Implementation of `ProblemTimeInterface::initTimestep()` . |

`closeTimestep` |
Implementation of `ProblemTimeInterface::closeTimestep()` . |

`problemStat` |
Returns the stored stationary problem. |

`oldSolutionVector` |
Returns the data vector for the old-solution |

`oldSolution` |
Returns the old-solution as discrete function |

`transferInitialSolution` |
Implementation of `ProblemTimeInterface::transferInitialSolution()` . |

## class `StandardProblemIteration`

Defined in header `<amdis/StandardProblemIteration.hpp>`

```
class StandardProblemIteration
: public virtual ProblemIterationInterface
```

A master problem for a single non-coupled problem.

### Member Functions

Function | Descriptions |
---|---|

(constructor) |
Constructs a StandardProblemIteration |

`beginIteration` |
Implementation of `ProblemIterationIterface::beginIteration()` |

`oneIteration` |
Implementation of ProblemIterationInterface::oneIteration() |

`endIteration` |
Implementation of ProblemIterationInterface::endIteration() |

`name` |
Returns the name of the iteration interface. |

`numProblems` |
Returns number of managed problems. |

`problem` |
Returns a managed `ProblemStat` . |

## class `ProblemStatBase`

Defined in header `<amdis/ProblemStatBase.hpp>`

```
class ProblemStatBase
```

Interface for stationary problems. Concrete problems must override all pure virtual methods. Base class for ProblemStat.

### Member Functions

Function | Descriptions |
---|---|

`markElements` |
Marks mesh elements for refinement and coarsening. |

`buildAfterAdapt` |
Assembling of system matrices and vectors after adaption. |

`adaptGrid` |
Refinement/coarsening of the grid. |

`globalCoarsen` |
Uniform global grid coarsening. |

`globalRefine` |
Uniform global grid refinement. |

`solve` |
Solves the assembled system. |

`estimate` |
A-posteriori error estimation. |

`name` |
Returns the name of the problem. |

## class `ProblemInstatBase`

Defined in header `<amdis/ProblemInstatBase.hpp>`

```
class ProblemInstatBase
: public virtual ProblemTimeInterface
```

Base class for ProblemInstat.

### Member Functions

Function | Descriptions |
---|---|

(constructor) |
Constructs a new `ProblemStatBase` |

`solveInitialProblem` |
Implementation of `ProblemTimeInterface::solveInitialProblem()` . |

`name` |
Returns the name of the instationary problem. |

`time` |
Returns reference to current simulation time set in `setTime()` from `AdaptInfo::time()` . |

`setTime` |
Implementation of `ProblemTimeInterface::setTime()` . |

`tau` |
Returns reference to current simulation timestep set in `setTime()` from `AdaptInfo::timestep()` . |

`invTau` |
Returns reference to current simulation `1.0/timestep` . |

## class `ProblemIterationInterface`

Defined in header `<amdis/ProblemIterationInterface.hpp>`

```
class ProblemIterationInterface
```

Interface for master problems needed by the adaption loop. A master problem
can handle one single or multiple coupled problems. In the latter case,
the master problem can determine the execution order of the build, solve,
estimate, and adapt steps of the single problems in `oneIteration()`

.

### Member Functions

Function | Descriptions |
---|---|

`beginIteration` |
Called before each adaption loop iteration. |

`oneIteration` |
Determines the execution order of the single adaption steps. |

`endIteration` |
Called after each adaption loop iteration. |

`numProblems` |
Returns number of managed problems. |

`problem` |
Returns the problem with the given number. |

`name` |
Returns the name of the iteration interface. |

## class `ProblemTimeInterface`

Defined in header `<amdis/ProblemTimeInterface.hpp>`

```
class ProblemTimeInterface
```

Interface for time dependent problems.

### Member Functions

Function | Descriptions |
---|---|

`initTimeInterface` |
Called at the beginning of the adaption loop before any other call. |

`setTime` |
Executes all needed operations when the simulation time changes. |

`initTimestep` |
Called at the beginning of each timestep. |

`closeTimestep` |
Called at the end of each timestep. |

`solveInitialProblem` |
Solves the initial problem. |

`transferInitialSolution` |
Transfer the initial problem. |