Solver

struct SGFEVector{T, Tv, Ti, ONBType<:ONBasis, MIType}

A structure that extends ExtendableFEMBase.FEVector to include information about the stochastic discretization, specifically the associated TensorizedBasis.

Fields

• FES_space: Array of finite element spaces for all stochastic modes.
• TB: The tensorized basis used for the stochastic discretization.
• active_modes: Indices of the active stochastic modes (with respect to TB.multi_indices).
• length4modes: Offsets for each mode in the global vector.
• FEVectorBlocks: Vector block mask for each multi-index (stochastic mode).
• entries: The full coefficient vector containing all degrees of freedom.
• last_sample: The last sample used for evaluation (for efficient repeated evaluation).
• FEV: An FEVector used for evaluating the SGFEVector at a given sample.

This structure enables efficient storage, evaluation, and manipulation of solutions in stochastic Galerkin finite element methods.

SGFEVector(
    FES::Array{<:ExtendableFEMBase.FESpace{Tv, Ti}, 1},
    TB::TensorizedBasis{Tv, ONBType, MIType};
    active_modes,
    T,
    unames
) -> SGFEVector

Constructs an SGFEVector for the given spatial finite element spaces FES and the tensorized basis TB representing the stochastic discretization.

Arguments

• FES: Array of finite element spaces (one for each unknown).
• TB: The tensorized basis for the stochastic discretization.
• active_modes: Indices of the active stochastic modes to include (default: all modes in TB).
• T: The floating-point type for the coefficient vector (default: Tv).
• unames: Names or identifiers for the unknowns (default: 1:length(FES)).

Returns

An SGFEVector object that stores the spatial and stochastic discretization, the coefficient vector, and all necessary metadata for efficient evaluation and manipulation in stochastic Galerkin FEM.

getindex(
    SGFEV::SGFEVector,
    u::Int64,
    i::Int64
) -> ExtendableFEMBase.FEVectorBlock

returns the FEVectorBlock for the i-th stochastic mode of the u-th unknown

getindex(
    SGFEV::SGFEVector,
    i::Int64
) -> ExtendableFEMBase.FEVectorBlock

returns the i-th stochastic mode

length(SGFEV::SGFEVector) -> Int64

returns the length of the full vector, i.e., the total number of degrees of freedom

show(io::IO, SGFEV::SGFEVector)

Custom show function for FEVector that prints some information on its blocks.

size(SGFEV::SGFEVector)

returns a tuple with the number of active stochastic modes and the number of spatial degrees of freedom

fetypes(SGFEV::SGFEVector) -> Vector

returns the finite element types of the (spatial) FE spaces

num_multiindices(SGFEV::SGFEVector) -> Int64

returns the number of active modes (that are used from the stored tensorized basis)

set_sample!(SGFEV::SGFEVector, S::AbstractVector)

Evaluates the SGFEVector at the given sample S and stores the result in SGFEV.FEV.

Arguments

• SGFEV: The stochastic Galerkin finite element vector to evaluate.
• S: A vector representing the sample (values for the stochastic variables).

Details

• The sample S is stored in SGFEV.last_sample (truncated or padded as needed).
• The tensorized basis is evaluated at S, and the resulting coefficients are used to assemble the spatial solution in SGFEV.FEV.
• This enables efficient evaluation of the SGFEM solution at arbitrary points in the stochastic parameter space.

solve!(
    ::Type{ExtendableASGFEM.AbstractModelProblem},
    sol::SGFEVector,
    C::AbstractStochasticCoefficient;
    rhs,
    use_iterative_solver,
    bonus_quadorder_f,
    bonus_quadorder_a,
    kwargs...
)

Solves the specified model problem using the provided stochastic coefficient C and right-hand side rhs, writing the solution into sol.

The sol vector communicates both the spatial and stochastic discretization, as well as any initial data required for the iterative solver.

• If use_iterative_solver (default: true) is set, an iterative solver is used. Otherwise, the full system matrix is assembled and solved directly (note: this is very slow for large systems).
• The parameters bonus_quadorder_f (default: 0) and bonus_quadorder_a (default: 2) allow you to increase the quadrature order for terms involving the right-hand side or the stochastic coefficient, respectively.
• Additional keyword arguments can be passed via kwargs.

If no solver is implemented for the given model problem, an error is thrown.

MyPreconditionerPrimal{Tv, FAC}

Block-diagonal LU preconditioner for the primal SG Poisson system.

Each diagonal block is the inverse of the mean stiffness matrix $A_0$, obtained via LU factorisation. Boundary dofs are handled by stiffening those diagonal entries in the pre-factored $A_0$ so that forward/backward substitution implicitly enforces zero Dirichlet conditions.

Fields

• LUA::FAC: LU factorisation of $A_0$ with stiffened boundary dofs.
• DA::Vector{Tv}: Diagonal of $A_0$ (kept for compatibility, not used in current path).
• bdofs::Vector{Int}: Boundary dofs.
• nmodes::Int: Number of stochastic modes.

MyPreconditionerPrimal(A::ExtendableSparseMatrix, bdofs, nmodes)

Construct a block-diagonal LU preconditioner from the mean matrix $A_0$.

Stiffens boundary diagonal entries to $1e60$ and precomputes the LU factorisation of the modified matrix. The factorisation is reused for every preconditioner-vector product in the GMRES iteration.

MySystemPrimal{Tv, MT, VT, GT}

Matrix-free block system evaluator for the primal stochastic Poisson problem.

Represents the block-structured linear operator arising from the stochastic Galerkin discretisation with a Karhunen-Loeve expanded diffusion coefficient. The operator acts on vectors partitioned by stochastic mode and implements mul! for matrix-vector products without assembling the full block matrix.

Fields

• A0::MT: Mean diffusion block.
• Am::Vector{MT}: KL perturbation blocks (one per random dimension).
• G::GT: Coupling tensor with entries G[e, mu, nu] mapping KL indices e to mode pairs.
• bdofs::Vector{Int}: Boundary dofs for homogeneous Dirichlet conditions.
• nmodes::Int: Number of stochastic modes.

solve_full_primal!(SolutionSGFEM::SGFEVector, A0, A, b, G, nmodes, rhsfac)

Build the full block SG Poisson matrix and solve it with a direct backslash.

Assembles the complete block system into a $FEMatrix$ with $nmodes * nmodes$ blocks: the diagonal receives contributions from the mean operator $A_0$, and off-diagonal and diagonal blocks are filled by the KL perturbations $A_e$ weighted by the coupling tensor $G$. Boundary dofs are stiffened and RHS entries set to zero. The system is solved in a single line via Julia's $\$ operator.

This function is primarily useful for verification against the matrix-free solver $solve_primal!$ since it scales as $O((nmodes * ndofs)^3)$.

Arguments

• SolutionSGFEM: Output $SGFEVector$ (modified in-place).
• A0: Mean stiffness block.
• A: Vector of KL-perturbation blocks.
• b: Right-hand side blocks.
• G: Coupling tensor.
• nmodes: Number of stochastic modes.
• rhsfac: Not used.

solve_primal!(SolutionSGFEM::SGFEVector, A0, Am, b0, G, nmodes, bfac; atol, rtol)

Solve the primal SG Poisson system iteratively using matrix-free preconditioned GMRES.

Builds a MySystemPrimal operator and a MyPreconditionerPrimal from the mean matrix $A_0$, KL perturbation blocks $A_m$, coupling tensor $G$, and boundary info extracted from the $SGFEVector$. The right-hand side is assembled by copying the solution vector, adding the deterministic force block $b_0$ to mode 1, and zeroing boundary entries on all modes.

Krylov.jl's $Krylov.gmres$ is used as the solver. Default tolerances are 1e-14.

Arguments

• SolutionSGFEM: Output $SGFEVector$ (modified in-place).
• A0: Mean stiffness block.
• Am: Vector of KL-perturbation stiffness blocks.
• b0: Deterministic right-hand side block (applied to mode 1 only).
• G: Coupling tensor $G[e, mu, nu]$.
• nmodes: Number of stochastic modes.
• bfac: Not used.

Keywords

• atol: Absolute GMRES tolerance (default 1.0e-14).
• rtol: Relative GMRES tolerance (default 1.0e-14).

LinearAlgebra.mul!(Ax, S::MySystemPrimal, x)

Matrix-free matmul for the block-structured SG Poisson operator.

Computes the product $Ax$ in-place without assembling the full block matrix. For each stochastic mode $\mu$, the deterministic diffusion $A_0$ is applied to mode $\mu$ of $x$, and every KL perturbation $A_e$ is applied to mode $\nu$ of $x$ weighted by the coupling coefficient $G_{e,\mu,\nu}$. Boundary rows are zeroed out after accumulation.