Taiga.FastDiagonalization(model::Poisson{Dim,T}; method::Type{<:DataApproximationMethod}=Wachspress, kwargs...) where {Dim,T}

Constructs a FastDiagonalization preconditioner using a particular data approximation method. Poisson supports Wachspress and ModalSplines data approximation methods.

Taiga.FastDiagonalization(::Type{<:ModalSplines}, model::Poisson; spaces::NTuple{Dim, SplineSpace{T}})

Constructs a fast diagonalization preconditioner using rank 1 modal splines approximation of model patch data on univariate spline spaces in spaces.

Taiga.FastDiagonalization(::Type{<:Wachspress}, model::Poisson; niter::Int)

Constructs a fast diagonalization preconditioner using niter of Wachspress algorithm to approximate patch data.

LinearOperator{Dim,T,M<:AbstractMatrix{T},V<:AbstractVector{T},B<:KroneckerProduct{T}} <: Taiga.LinearOperator{Dim,T}

Linear operator for the homogeneous system in Poisson model.

Fields:

• K::M: stiffness matrix
• b::V: rhs vector
• C::B: Kronecker product extraction operator

LinearOperatorApproximation{Dim,T} <: Taiga.LinearOperatorApproximation{Dim,T}

Linear operator approximation for the homogeneous system in Poisson model.

Fields:

• L::KroneckerProductAggregate{T}: Kronecker product aggregate

MatrixfreeLinearOperator{Dim, T, M <: Poisson{Dim, T}, A <: Accessor{Dim}, B <: KroneckerProduct{T}, V <: VectorSumfactoryCache{Dim}} <: Taiga.LinearOperator{Dim,T}

Matrix-free linear operator for the homogeneous system in Poisson model.

Fields:

• model::M: model
• acc::A: patch accessor
• C::B: Kronecker product extraction operator
• pullback_body::Function: pullback for the body terms
• sumfact_cache_u::V: sum factorization vector cache

Poisson{Dim,T,M<:GeometricMapping{Dim},V<:ScalarSplineSpace{Dim,T},W<:Field{Dim}} <: Model{Dim,T}

Strong form

-∇ ⋅ (κ∇u) = f      in Ω
         u = ū      on Γ₁
    ∇u ⋅ n = t ⋅ n  on Γ₂

Weak form

Find the homogeneous solution uₒ ∈ Sʰ, s.t. ∀v ∈ Sʰ

    ∫ ∇(uₒ - ū) ⋅ ∇v dΩ - ∫ (t ⋅ n) v dΓ₁ = - ∫ f v dΓ

for the homogeneous problem uₒ = 0 on Γ. The solution
u ∈ ū ⊕ Sʰ satisfying nonhomogeneous Dirichlet boundary
conditions is u = uₒ + ū.

Note

The linear operators Taiga.PoissonModule.LinearOperator and Taiga.PoissonModule.MatrixfreeLinearOperator correspond to the homogeneous problem on the constrained space. The solution of the nonhomogeneous problem is obtained by Taiga.PoissonModule.apply_particular_solution.

Fields:

• F::M: geometric mapping
• S::V: solution space
• Δ::Partition{Dim,T}: partition of the space
• uʰ::Field: solution field
• ūʰ::W: scalar solution field satisfying Dirichlet boundary conditions
• κ::Function: conductivity matrix of size Dim × Dim as function of spatial coordinates
• f::Function: body force as function of spatial coordinates
• t::Function: traction as function of spatial coordinates

Poisson(F::M, S::V, C::ScalarSplineSpaceConstraints{Dim}, uʰ::W, ūʰ::W, κ::Function, f::Function, t::Function; matrixfree=false) where {Dim,T,M<:GeometricMapping{Dim},V<:ScalarFunctionSpace{Dim,T},W<:Field{Dim}}

Construct Poisson model.

Arguments:

• F::M: geometric mapping
• S::V: scalar function space
• C::ScalarSplineSpaceConstraints{Dim}: function space constraints
• uʰ::W: scalar solution field
• ūʰ::W: scalar solution field satisfying Dirichlet boundary conditions
• κ::Function: conductivity matrix as a function of spatial coordinates
• f::Function: body force as function of spatial coordinates
• t::Function: traction vector as function of spatial coordinates
• matrixfree::Bool: boolean flag for matrix-free forward problem, default is false

apply_particular_solution(model::Poisson, x₀::V) where {V<:AbstractVector}

Applies particular solution to homogeneous solution vector.

apply_particular_solution(L::LinearOperator, x₀::V) where {V<:AbstractVector}

Applies particular solution to homogeneous solution vector.

forcing!(b::AbstractVector, L::LinearOperator)

Updates forcing vector b.

forcing(L::LinearOperator)

Returns forcing vector.

linear_operator(model::Poisson; matrixfree::Bool=false, constrained::Bool=true)

Returns a linear operator corresponding for model. Depending on the matrixfree keyword either PoissonModule.LinearOperator or PoissonModule.MatrixfreeLinearOperator is returned. For PoissonModule.MatrixfreeLinearOperator the constrained flag controls if the extraction operator is built into the matrix-free linear operator.

linear_operator_approximation(model::Poisson; method::Type{<:DataApproximationMethod})

Returns PoissonModule.LinearOperatorApproximation for Poisson model using one of Taiga.DataApproximationMethod.