Finite element spaces

Abstract type to represent an finite-element space of size S. See SingleFESpace for more details about what looks like a finite-element space.

Devs notes

All subtypes should implement the following functions:

• get_function_space(feSpace::AbstractFESpace)
• get_shape_functions(feSpace::AbstractFESpace, shape::AbstractShape)
• get_cell_shape_functions(feSpace::AbstractFESpace, shape::AbstractShape)
• get_ndofs(feSpace::AbstractFESpace)
• is_continuous(feSpace::AbstractFESpace)

Alternatively, you may define a "parent" to your structure by implementing the Base.parent function. Then, all the above functions will be redirected to the "parent" FESpace.

Devs notes

All subtypes should implement the following functions:

• get_fespace(mfeSpace::AbstractMultiFESpace)
• get_mapping(mfeSpace::AbstractMultiFESpace)
• get_dofs(mfeSpace::AbstractMultiFESpace, icell::Int)
• get_shape_functions(mfeSpace::AbstractMultiFESpace, shape::AbstractShape)
• get_cell_shape_functions(mfeSpace::AbstractMultiFESpace, shape::AbstractShape)

A MultiFESpace represents a "set" of TrialFESpace or TestFESpace. This structure provides a global dof numbering for each FESpace.

N is the number of FESpace contained in this MultiFESpace.

Note that the FESpace can be different from each other (one continous, one discontinuous; one scalar, one vector...)

MultiFESpace(
    feSpaces::Tuple{Vararg{TrialOrTest, N}};
    arrayOfStruct::Bool = AOS_DEFAULT,
) where {N}
MultiFESpace(
    feSpaces::AbstractArray{FE};
    arrayOfStruct::Bool = AOS_DEFAULT,
) where {FE <: TrialOrTest}
MultiFESpace(feSpaces::Vararg{TrialOrTest}; arrayOfStruct::Bool = AOS_DEFAULT)

Build a finite element space representing several sub- finite element spaces.

This is particulary handy when several variables are in play since it provides a global dof numbering (for the whole system). The finite element spaces composing the MultiFESpace can be different from each other (some continuous, some discontinuous, some scalar, some vectors...).

Arguments

• feSpaces : the finite element spaces composing the MultiFESpace. Note that they must be of type TrialFESpace or TestFESpace.

Keywords

• arrayOfStruct::Bool = AOS_DEFAULT : indicates if the dof numbering should be of type "Array of Structs" (AoS) or "Struct of Arrays" (SoA).

SingleFESpace(
    fSpace::AbstractFunctionSpace,
    mesh::AbstractMesh,
    dirichletBndNames = String[];
    size::Int = 1,
    isContinuous::Bool = true,
    kwargs...
)

Build a finite element space (scalar or vector) from a FunctionSpace and a Mesh.

Arguments

• fSpace::AbstractFunctionSpace : the function space associated to the FESpace
• mesh::AbstractMesh : the mesh on which the FESpace is discretized
• dirichletBndNames = String[] : list of mesh boundary labels where a Dirichlet condition applies

Keywords

• size::Int = 1 : the number of components of the FESpace
• isContinuous::Bool = true : if true, a continuous dof numbering is created. Otherwise, dof lying

on cell nodes or cell faces are duplicated, not shared (discontinuous dof numbering)

• kwargs : for things such as parallel cache (internal/dev usage only)

An finite-element space (FESpace) is basically a function space, associated to degrees of freedom (on a mesh).

A FESpace can be either scalar (to represent a Temperature for instance) or vector (to represent a Velocity). In case of a "vector" SingleFESpace, all the components necessarily share the same FunctionSpace.

TestFESpace(trialFESpace::TrialFESpace)
TestFESpace(
    fSpace::AbstractFunctionSpace,
    mesh::AbstractMesh,
    dirichletBndNames = String[];
    size::Int = 1,
    isContinuous::Bool = true,
    kwargs...,
)

Build a test finite element space.

A TestFESpace can be built from a TrialFESpace. See SingleFESpace for hints about the function arguments. Only arguments specific to TrialFESpace are detailed below.

Examples

julia> mesh = one_cell_mesh(:line)
julia> fSpace = FunctionSpace(:Lagrange, 2)
julia> U = TrialFESpace(fSpace, mesh)
julia> V = TestFESpace(U)

A TestFESpace is basically a SingleFESpace plus other attributes (related to boundary conditions)

TrialFESpace(feSpace, dirichletValues)
TrialFESpace(
    fSpace::AbstractFunctionSpace,
    mesh::AbstractMesh,
    dirichlet::Dict{String} = Dict{String, Any}();
    size::Int = 1,
    isContinuous::Bool = true,
    kwargs...
)
TrialFESpace(
    fSpace::AbstractFunctionSpace,
    mesh::AbstractMesh,
    type::Symbol,
    dirichlet::Dict{String} = Dict{String, Any}();
    size::Int = 1,
    kwargs...
)

Build a trial finite element space.

See SingleFESpace for hints about the function arguments. Only arguments specific to TrialFESpace are detailed below.

Arguments

• dirichlet::Dict{String} = Dict{String, Any}() : dictionnary specifying the Dirichlet valued-function (or function) associated to each mesh boundary label. The function f(x,t) to apply is expressed in the physical coordinate system. Alternatively, a constant value can be provided instead of a function.
• type::Symbol : :continuous or :discontinuous

Warning

For now the Dirichlet condition can only be applied to nodal bases.

Examples

julia> mesh = one_cell_mesh(:line)
julia> fSpace = FunctionSpace(:Lagrange, 2)
julia> U = TrialFESpace(fSpace, mesh)
julia> V = TrialFESpace(fSpace, mesh, :discontinuous; size = 3)
julia> W = TrialFESpace(fSpace, mesh, Dict("North" => 3., "South" => (x,t) -> t .* x))

A TrialFESpace is basically a SingleFESpace plus other attributes (related to boundary conditions)

Dev notes

• we cannot directly store Dirichlet values on dofs because the Dirichlet values needs "time" to apply

A MultiplierFESpace can be viewed as a set of independent P0 elements. It is used to define Lagrange multipliers and assemble the associated augmented system (the system that adds the multipliers as unknowns).

Low-level constructor

Build a global numbering using an Array-Of-Struct strategy

Build a global numbering using an Struct-Of-Array strategy

allocate_dofs(feSpace::AbstractFESpace, T = Float64)

Allocate a vector with a size equal to the number of dof of the FESpace, with the type T. For a MultiFESpace, a vector of the total size of the space is returned (and not a Tuple of vectors)

build_jacobian_sparsity_pattern(u::AbstractMultiFESpace, mesh::AbstractMesh)

Build the jacobian sparsity pattern corresponding to the "FESpace U".

A jacobian is associated to a given function, here this function is assumed to mix (i.e "link") all the variables of the FESpace together.

Return the shape functions associated to the AbstractFESpace in "packed" form: λ(x) = (λ₁(x),...,λᵢ(x),...λₙ(x)) for the n dofs.

Return the boundary tags where a Dirichlet condition applies

Return the values associated to a Dirichlet condition

Return the dofs indices for the cell icell

Result is an array of integers.

get_dofs(feSpace::MultiFESpace, icell::Int)

Return the dofs indices for the cell icell for each single-feSpace. Result is a tuple of array of integers, where each array of integers are the indices relative to the numbering of each singleFESpace.

Warning:

Combine get_dofs with get_mapping if global dofs indices are needed.

get_fespace(mfeSpace::AbstractMultiFESpace, iSpace)
get_fespace(mfeSpace::AbstractMultiFESpace)

Return the i-th FESpace composing this AbstractMultiFESpace. If no index is provided, the tuple of FESpace composing this AbstractMultiFESpace` is returnted.

Return the tuple of FESpace composing this MultiFESpace

Return the FunctionSpace (eventually multiple spaces) associated to the AbstractFESpace.

get_mapping(mfeSpace::AbstractMultiFESpace, iSpace)
get_mapping(mfeSpace::AbstractMultiFESpace)

Return the mapping for the ith FESpace composing the MultiFESpace. If no index is provided, the tuple of mapping for each FESpace` is returnted.

Number of FESpace composing the MultiFESpace

Return the size S(= number of components) associated to AbstractFESpace{S}.

Return the total number of dofs of the FESpace, taking into account the continuous/discontinuous type of the space. If the FESpace contains itself several FESpace (see MultiFESpace), the sum of all dofs is returned.

Total number of dofs contained in this MultiFESpace

Return the shape functions associated to the AbstractFESpace.

Return the size S associated to AbstractFESpace{S}.