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The `FL.DISCRETIZATION`

package defines
`<discretization>`

as an abstract class and
`<fe-discretization>`

as a concrete derived class.

The key for local assembly is given by the generic function
`get-fe`

, which yields a suitable finite element for a given cell.
The value of `get-fe`

is a class `<fe>`

for scalar problems
or `<vector-fe>`

for vector-valued problems which contains
information on base functions and node functionals. Another generic
function `quadrature-rule`

computes memoized quadrature rules for
those finite elements.

Obviously, also non-standard finite element discretizations like hp-methods
would fit into this scheme. The key for local assembly is given by the
generic function `get-fe`

, which yields a suitable finite element
for a given cell. The value of `get-fe`

is a class `<fe>`

for scalar problems or `<vector-fe>`

for vector-valued problems which
contains information on base functions and node functionals. Another
generic function `quadrature-rule`

computes memoized quadrature
rules for those finite elements.

The file `lagrange.lisp` provides Lagrange finite elements of
arbitrary order. The evaluation points for the node functionals may be
chosen either uniformly distributed or at the Gauss-Lobatto points. The
latter choice of points yields better stability properties but is
restricted to cube meshes. Also functions for constructing cell mappings
by pointwise evaluation of the domain boundary are provided here, which may
be used to construct isoparametric domain approximations.

In the file `fedisc.lisp`, the function `fe-discretize`

is
defined. This function performs the standard steps for finite element
discretization: interior assembly, assembly and elimination of hanging-node
and essential-boundary constraints. It works on a blackboard as explained
in Section Blackboards and can reuse already available matrix-vector
structure. There is a somewhat less flexible interface provided by the
funtion `discretize-globally`

which calls
`fe-discretize`

.

In the files `ellsys-fe.lisp`, `elasticity-fe.lisp` and
`navier-stokes.lisp` one can find methods for local assembly for the
different problems. They are defined in own packages which use both the
package `FL.DISCRETIZATION`

and the package for the particular
problem.

- Variable:
***QUADRATURE-ORDER*** -
Quadrature order to be used. NIL determines the order automatically.

- Variable:
***SUGGESTED-DISCRETIZATION-ORDER*** -
The suggested order of discretization. In non-nil, this value should be taken into account by methods to

`select-discretization`.

- Class:
**<ANSATZ-SPACE-AUTOMORPHISM>** -
A automorphism of an ansatz space.

Superclasses: <ANSATZ-SPACE-MORPHISM> <ANSATZ-SPACE-OBJECT>

- Class:
**<ANSATZ-SPACE-MORPHISM>** -
A mapping between two ansatz-spaces.

- Class:
**<ANSATZ-SPACE-VECTOR>** -
A sparse vector which is interpreted as the ansatz-space for a specific fe-class on a given mesh.

Superclasses: <ANSATZ-SPACE-OBJECT> <HT-SPARSE-VECTOR>

- Class:
**<ANSATZ-SPACE>** -
A finite element ansatz space is determined by finite element discretization, mesh and problem. The constraints are stored in the slot

`properties`.Superclasses: PROPERTY-MIXIN

Direct slots:

- FE-CLASS: The finite element class for this ansatz space.
- PROBLEM: The proplem for this ansatz space which determines essential constraints.
- MESH: The mesh for this ansatz space which determines hanging-node constraints.

- Class:
**<DISCRETIZATION>** -
Discretization base class.

Superclasses: PROPERTY-MIXIN

- Class:
**<FE-DISCRETIZATION>** -
FE discretization base class.

Superclasses: <DISCRETIZATION>

- Class:
**<SCALAR-FE-DISCRETIZATION>** -
Class for scalar fe discretizations.

Superclasses: <STANDARD-FE-DISCRETIZATION>

Direct slots:

- ORDER: Self-explanatory.

- Class:
**<SCALAR-FE>** -
A finite element <fe> is given for each reference cell, e.g. <2-simplex>. dofs are the degrees of freedom associated with the cell, basis is the dual basis to dofs in some polynomial space. subcell-ndofs is the number of ndofs on each subcell. subcell-indices is a list of indices for all subcells with dofs. Usually, the <scalar-fe> will occur as values of a procedure or as values in a hash-table with the reference cells as keys.

Superclasses: <FE>

Direct slots:

- DOFS: The dofs which are functionals satisfying dof[i](basis[j])=delta(i,j)$.
- BASIS: The fe’s basis functions.
- ORDER: Self-explanatory.

- Class:
**<STANDARD-FE-DISCRETIZATION>** -
Finite element discretization where the finite element depends only on the cell (usually via its reference cell).

Superclasses: <FE-DISCRETIZATION>

Direct slots:

- CELL->FE: A function mapping a cell to a finite element. Usually this will be a closure memoized over the reference cell.

- Class:
**<VECTOR-FE-DISCRETIZATION>** -
Vector FE discretization class.

Superclasses: <STANDARD-FE-DISCRETIZATION>

Direct slots:

- COMPONENTS: Either a vector of components or the symbol :variable, if the components vary across the domain as is the case for hp-FEM.

- Class:
**<VECTOR-FE>** -
Finite element for vector functions. Components is a vector of scalar finite elements. Subcell-offsets is an array consisting of arrays which yield such an offset for every subcell.

Superclasses: <FE>

Direct slots:

- COMPONENTS: Self-explanatory.
- DOFS: Self-explanatory.
- BASIS: Self-explanatory.

- Function:
**ASSEMBLE-INTERIOR**`ANSATZ-SPACE``WHERE`&KEY`LEVEL``MATRIX``MASS-MATRIX``RHS``PARALLEL-CLUSTERING`&ALLOW-OTHER-KEYS -
Assemble the interior, i.e. ignore constraints arising from boundaries and hanging nodes. Discretization is done using the ansatz space

`ansatz-space`on level`level`. The level argument will usually be`NIL`

when performing a global assembly, and be equal to some number when assembling coarse level matrices for multigrid. The argument`where`is a flag indicating where assembly is to be done. It should be one of the keywords`:surface`

,`:refined`

,`:all`

. The arguments`matrix`,`rhs`should contain vectors/matrices where the local assembly is accumulated. Boundary conditions and constraints are not taken into account within this routine.In general, this function does most of the assembly work. Other steps like handling constraints are intricate, but usually of lower computational complexity.

- Function:
**CELL-INTEGRATE**`CELL``OBJECT`&KEY`INITIAL-VALUE``COMBINER``KEY``COEFF-FUNC``NO-WEIGHTS-P``QUADRATURE-ORDER`&ALLOW-OTHER-KEYS -
Integrates

`object`on`cell`.

- Function:
**CELL-KEY**`CELL``MESH` -
If cell is identified, its identification is the key.

- Function:
**CELL-LAGRANGE-FE**`CELL``ORDER``TYPE`&OPTIONAL`DISC` -
Returns a Lagrange fe depending on reference cell, an order (which can be number or vector), and a type symbol.

- Function:
**CHOOSE-START-VECTOR**`ANSATZ-SPACE` -
Choose a reasonable start vector for some strategy.

- Function:
**COMPONENT**`FE``COMP` -
Returns the

`comp`-th component of the (vector) finite element`fe`.

- Function:
**CONSTRAINED-INTERPOLATION-MATRIX**`ANSATZ-SPACE`&KEY`LEVEL``WHERE``IMAT`(`TYPE``LOCAL`) -
The multigrid algorithm needs an interpolation which satisfies the constraints like essential or periodic boundary conditions.

- Function:
**CONSTRUCT-COEFF-INPUT**`CELL``GLOBAL``DPHI``VALUES``GRADIENTS``FE-PARAMETERS` -
Constructs a coefficient input list from FE data

`cell`is the cell,`global`is the global coordinate of the integration point,`values`and`gradients`the values and gradients of the shape functions at the ip, and`fe-parameters`are the corresponding data of fe-functions to be evalutated.

- Function:
**DISCRETIZATION-ORDER**`DISC` -
Returns the order of the discretization.

- Function:
**DISCRETIZE**`DISCRETIZATION``PROBLEM``BLACKBOARD` -
Calculates matrix and right-hand side for the

`discretization`and the`problem`. Further parameters are provided inside the`blackboard`.

- Function:
**DISCRETIZE-GLOBALLY**`PROBLEM``H-MESH``FE-CLASS` -
Discretize

`problem`on the hierarchical mesh`h-mesh`using finite elments given by`fe-class`.

- Function:
**DISCRETIZE-LOCALLY**`PROBLEM``COEFFS``FE``QRULE``FE-GEOMETRY`&KEY`MATRIX``MASS-MATRIX``RHS``LOCAL-U``LOCAL-V``FE-PARAMETERS``RESIDUAL-P`&ALLOW-OTHER-KEYS -
Computes a local stiffness matrix and right-hand side. The algorithm will usually work as follows:

- Get coefficient functions for the patch of the cell.
- Compute geometry information for all ips (values and gradients of the shape functions).
- Loop over integration points ip:
- If necessary, compute input for coefficient functions.
This input may contain values of finite element function in the
property list
`fe-parameters`. - Evaluate coefficient functions at ips.
- Add the contributions for matrix and right-hand side to
`local-mat`and`local-rhs`.

- If necessary, compute input for coefficient functions.
This input may contain values of finite element function in the
property list

If

`local-u`and`local-v`are set, then`local-v`*`local-mat`*`local-u`and`local-v`*`local-rhs`is computed. This feature may be used later on for implementing matrixless computations.

- Class:
**DOF** -
Degree of freedom in a finite element. It is defined as a functional defined by integration over a sub-cell or by evaluation at a local coordinate of a sub-cell of a reference cell.

Direct slots:

- INDEX: Index of the dof in the cell-dof array
- SUBCELL: Reference subcell on which the dof is defined
- SUBCELL-INDEX: Index of the reference subcell on which the dof is defined
- IN-VBLOCK-INDEX: Index of the dof in the subcell vblock
- PRECISE-COORD: Local coordinate of the dof in the reference subcell in high precision.
- COORD: Local coordinate of the dof in the reference subcell
- PRECISE-GCOORD: Global coordinate of the dof on the reference cell in high precision.
- GCOORD: Global coordinate of the dof on the reference cell. This is a vector in a standard floating point precision which is sufficient for function evaluation, but not sufficient for calculating stiffness matrix entries.
- FUNCTIONAL: A functional for functions defined on the reference cell

- Function:
**ELIMINATE-CONSTRAINTS**`MAT``RHS``CONSTRAINTS-P``CONSTRAINTS-Q``CONSTRAINTS-R`&KEY`ASSEMBLE-LOCALLY``INCLUDE-CONSTRAINTS` -
Constraints are given by an equation: P x = Q x + r

Here x in V, P is an orthogonal projection on a subspace V_P of V, Q maps some other space which may have nonempty intersection with V_P to V_P. With S we denote the mapping Id-P. This function returns the matrix for a Galerkin method on the constrained space. It is used for treating hanging nodes and essential boundary conditions. When assemble-locally is t the sparse structure of mat is used instead of the sparse structure of the hanging node interface. When include-constraints is non-nil, the constraints are included in matrix and rhs.

- Function:
**ESSENTIAL-BOUNDARY-CONSTRAINTS**`PROBLEM``ANSATZ-SPACE`&KEY`LEVEL``WHERE``INTERFACE` -
Computation of essential constraints. Should probably be incorporated into the ansatz-space definition.

- Function:
**EXTENDED-EXTRACT**`MAT``KEYS`&KEY`ROW?``COL?` -
Extract a sub-matrix from a sparse matrix for the given keys.

- Function:
**FE-CELL-GEOMETRY**`CELL``SAMPLE-POINTS`&KEY`WEIGHTS``METRIC``VOLUME`&AUX (`N`(`LENGTH``SAMPLE-POINTS`)) -
Collects cell geometry information at

`sample-points`inside a property list.

- Function:
**FE-DISCRETIZE**`BLACKBOARD` -
Finite element discretization for an ansatz space provided on the blackboard.

- Function:
**FE-EXTRACTION-INFORMATION**`FE``INDICES` -
Computes information for extracting components out of a vector finite element.

- Function:
**FE-EXTREME-VALUES**`ASV`&KEY`CELLS``SKELETON``COMPONENT` -
Computes the extreme values of a finite element function over the domain or some region. The result is a pair, the car being the minimum values and the cdr the maximum values. Each part is a matrix of the format ncomps x multiplicity.

- Function:
**FE-GRADIENT**`ASV``GLOBAL-POS` -
Evaluates the gradient of the FE ansatz-space-vector

`asv`at`global-pos`.

- Function:
**FE-INTEGRATE**`ASV`&KEY`CELLS``SKELETON``INITIAL-VALUE``COMBINER``KEY``COEFF-FUNC``NO-WEIGHTS-P` -
Integrates a finite element function over the domain. key is a transformer function, as always (e.g. #’abs if you want the L1-norm).

- Function:
**FE-LOCAL-GRADIENT**`ASV``CELL``LOCAL-POS` -
Evaluates the gradient of a FE ansatz-space-vector on

`cell`at`local-pos`.

- Function:
**FE-LOCAL-VALUE**`ASV``CELL``LOCAL-POS` -
Evaluates a FE ansatz-space-vector in

`cell`at`local-pos`.

- Function:
**FE-VALUE**`ASV``GLOBAL-POS` -
Evaluates a FE ansatz-space-vector at

`global-pos`.

- Function:
**GAUSS-RULE**`FACTOR-DIMS``S` -
Returns an s-point Gauss integration rule.

- Function:
**GET-FE**`FE-DISC``CELL` -
Returns the finite element for the given discretization and reference cell.

- Function:
**GET-LOCAL-FROM-GLOBAL-MAT**`GLOBAL-MAT``CELL`&OPTIONAL`DOMAIN-CELL` -
Maps the region in the global stiffness matrix determined by cell to a local matrix array.

- Function:
**GET-LOCAL-FROM-GLOBAL-VEC**`CELL``GLOBAL-VEC` -
Maps the region in global-vec determined by cell to a local vector.

- Function:
**HIERARCHICAL-MESH**`ASO` -
The hierarchical mesh for the given ansatz-space or ansatz-space object.

- Function:
**INCREMENT-GLOBAL-BY-LOCAL-MAT**`GLOBAL-MAT``LOCAL-MAT``CELL`&OPTIONAL`DOMAIN-CELL` -
Increments the region in global-mat determined by cell to the values of local-mat.

- Function:
**INCREMENT-GLOBAL-BY-LOCAL-VEC**`CELL``GLOBAL-VEC``LOCAL-VEC` -
Increments the region in global-vec determined by cell to the values of the local vector array.

- Function:
**INTEGRATION-RULE**`CELL``ORDER` -
Returns an integration rule for

`cell`of the given`order`.

- Function:
**INTERPOLATE-ON-REFCELL**`FE``FUNCTION` -
Interpolates

`function`on the reference cell of the finite element`fe`. Returns a standard-matrix corresponding to the block in the sparse vector.

- Function:
**INTERPOLATION-MATRIX**`ANSATZ-SPACE`&KEY`LEVEL``REGION``IMAT`(`TYPE``*INTERPOLATION-TYPE*`) -
On each cell of the skeleton

`region`or on all cells of level`level`of the mesh of`ansatz-space`, a local interpolation matrix is collected into an interpolation matrix.`type`is the interpolation type having a default value`*interpolation-type*`.

- Function:
**IP-GRADIENTS**`FE``OBJ` -
Returns a vector of local gradient matrices for

`obj`which may be a vector of integration points or a quadrature rule. Note that this function is memoized using an :around method.

- Function:
**IP-VALUES**`FE``OBJ` -
Returns a vector of ip values for

`obj`which may be a vector of integration points or a quadrature rule. Note that this function is memoized using an :around method.

- Function:
**LAGRANGE-ANSATZ-SPACE**`PROBLEM``MESH`&KEY (`ORDER`1) (`TYPE``UNIFORM`) -
A constructor for a problem-dependent Lagrange fe. Here, the number of components may vary with the respective patch.

- Function:
**LAGRANGE-FE**`ORDER`&KEY (`NR-COMPS`1) (`TYPE``UNIFORM`) -
Constructor for Lagrange fe. nr-comps=nil builds a scalar fe-discretization, otherwise a vector-fe-discretization is built.

- Function:
**LAGRANGE-MAPPING**`ORDER`&OPTIONAL (`TYPE``UNIFORM`) -
Returns a function which maps a cell by a polynomial which is obtained by interpolating the boundary map via Lagrange interpolation.

- Function:
**LOCAL-IMATRIX**`RULE``FE`&OPTIONAL (`TYPE``LOCAL`) -
Memoized call of compute-local-imatrix.

- Function:
**LOCAL-PMATRIX**`RULE``FE` -
Memoized call of compute-local-pmatrix.

- Function:
**LOCAL-TRANSFER-MATRIX**`FE-FROM``FE-TO` -
Computes a local transfer matrix between different FE spaces.

- Function:
**MAKE-FE-ANSATZ-SPACE**`FE-CLASS``PROBLEM``MESH` -
Constructor of

`<ansatz-space>`

.

- Function:
**MAKE-LOCAL-MAT**`AS1``CELL1`&OPTIONAL`AS2``CELL2` -
Generates a local matrix discretization for the given ansatz-space(s).

- Function:
**MAKE-LOCAL-VEC**`ANSATZ-SPACE``CELL`&KEY`MULTIPLICITY` -
Generates a local vector for local discretization. We allow overriding the default multiplicity for the ansatz space for special use cases.

- Function:
**MULTIPLE-EVALUATE-LOCAL-FE**`LOCAL-VEC``SHAPE-VALUES` -
Evaluates the vector given in

`local-vec`at multiple points. Here`local-vec`should be a data vector obtained with`get-local-from-global-vec`

and`ip-values`should be a vector obtained from`ip-values`

.

- Function:
**NR-OF-DOFS**`FE` -
Number of degrees of freedom for the (vector) finite element.

- Function:
**NR-OF-INNER-DOFS**`FE` -
Number of inner degrees of freedom for the (vector) finite element.

- Function:
**PRODUCT-RULE***&REST*`QUADRATURE-RULES` -
Computes a product rule for several lower-dimensional quadrature rules.

- Function:
**PROJECTION-MATRIX**`ANSATZ-SPACE`&KEY`LEVEL``REGION``PMAT` -
The algorithm works as follows: On each cell of the provided cell list or the whole refinement a local projection matrix computed on the reference finite element is copied into the global projection matrix.

- Function:
**QUADRATURE-RULE**`FE` -
Computes the quadrature rule to be used for the finite element

`fe`.

- Function:
**RANDOM-ANSATZ-SPACE-VECTOR**`ANSATZ-SPACE` -
Returns a ansatz space vector for

`ansatz-space`filled with random entries. Essential constraints are satisfied.

- Function:
**REPRESENTATIVE**`OBJ` -
Returns a representative for this object.

- Function:
**SELECT-DISCRETIZATION**`PROBLEM``BLACKBOARD` -
Select a discretization for the given

`problem`depending on the parameters on the`blackboard`.

- Function:
**SET-GLOBAL-TO-LOCAL-MAT**`GLOBAL-MAT``LOCAL-MAT``CELL`&OPTIONAL`DOMAIN-CELL` -
Sets the region in global-mat determined by cell to the values of the local matrix array.

- Function:
**SET-GLOBAL-TO-LOCAL-VEC**`CELL``GLOBAL-VEC``LOCAL-VEC` -
Sets the region in global-vec determined by cell to the values of the local vector array.

- Function:
**SPECIAL-ANSATZ-SPACE-VECTOR**`ANSATZ-SPACE`&OPTIONAL (`TYPE``RANDOM`) (`VALUE`1.0) (`CONSTRAINTS-P``T`) -
Returns a ansatz space vector for

`ansatz-space`filled with constant or random entries. Essential constraints are satisfied.

- Function:
**SUBCELL-OFFSETS**`FE` -
Reader for subcell-offsets. This is an array of length the number of components. Each component is an array giving the offset of this component in a sparse vector value block corresponding to the subcell.

- Function:
**TRANSFER-MATRIX**`DOMAIN-AS``IMAGE-AS`&KEY`NO-SLAVES` -
Builds a transfer matrix from domain-as to image-as.

- Function:
**WEIGHTS-FOR-IPS**`BETA``IPS` -
Determines weights for the integration points

`ips`such that they integrate*int_{-1}^{1} (1+y)^beta ... dy*with optimal order.