Model Problems
The following model problems are available and are dispatched via the following types.
ExtendableASGFEM.PoissonProblemPrimal — Typeabstract type PoissonProblemPrimal <: ExtendableASGFEM.AbstractModelProblemPoisson problem with linear stochastic coefficient $a$ that seeks $u$ such that
$-\mathrm{div}(a(y,x) \nabla u(y,x)) = f(x) \quad \text{for } (y,x) \in \Gamma \times D$
ExtendableASGFEM.LogTransformedPoissonProblemPrimal — Typeabstract type LogTransformedPoissonProblemPrimal <: ExtendableASGFEM.AbstractModelProblemPoisson problem with exponential stochastic coefficient $e^a$ that seeks $u$ such that
$-\mathrm{div}(e^a(y,x) \nabla u(y,x)) = f(x) \quad \text{for } (y,x) \in \Gamma \times D$
The stochastic Galerkin FEM solves the equivalent transformed problem
$-\mathrm{div}(\nabla u(y,x)) - \nabla a(y,x) \cdot \nabla u(y,x) = e^{-a(y,x)} f(x) \quad \text{for } (y,x) \in \Gamma \times D$
ExtendableASGFEM.LogTransformedPoissonProblemDual — Typeabstract type LogTransformedPoissonProblemDual <: ExtendableASGFEM.AbstractModelProblemPoisson problem with exponential stochastic coefficient $e^a$ that seeks $u$ such that
$-\mathrm{div}(e^a(y,x) \nabla u(y,x)) = f(x) \quad \text{for } (y,x) \in \Gamma \times D$
The dual formulation of the log-transformed formulation of the Poisson problem introduces the auxiliary stress variable $p := - e^a(y,x) \nabla u(y,x) = -∇ũ - ∇aũ$ for the transformed $ũ := e^{-a} u$.
Its weak formulation seeks $(p,ũ)$ such that
\[\begin{aligned} (p, q) + (ũ \nabla a, q) - (\mathrm{div}(q), ũ) & = 0 \quad \text{for all } q \\ -(\mathrm{div}(p), v) & = (f, v) \quad \text{for all } v \end{aligned}\]