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Script on Poisson Problems

The script scripts/poisson.jl provides an implementation of adaptive stochastic Galerkin FEM for various formulations of the parametric Poisson problem.

Available Problem Types

  • PoissonProblemPrimal: Standard Poisson problem with linear coefficient a
  • LogTransformedPoissonProblemPrimal: Log-transformed Poisson problem with exponential coefficient exp(a)
  • LogTransformedPoissonProblemDual: Dual formulation of the log-transformed problem (WIP)

Usage

The main function to run experiments is:

run(; problem = problem_type, kwargs...)

Results can be loaded and visualized using:

show_results(; kwargs...)    # show results and statistics
produce_plots(; kwargs...)   # generate convergence and other plots

Parameters

The following parameters can be specified (with their defaults):

Problem Configuration

  • problem = LogTransformedPoissonProblemPrimal: Problem type to solve
  • domain = "square": Domain shape ("square" or "lshape")
  • order = 1: Polynomial order of finite elements
  • initial_refs = 1: Initial uniform mesh refinements

Stochastic Parameters

  • C = StochasticCoefficientCosinus: Type of stochastic coefficient
  • decay = 2: Decay rate of the coefficient expansion
  • mean = 0: Mean value of the coefficient
  • maxm = 150: Maximum number of terms in coefficient expansion
  • initial_modes = [[0]]: Initial active stochastic modes

Adaptive Parameters

  • θ_stochastic = 0.5: Marking parameter for stochastic refinement
  • θ_spatial = 0.5: Marking parameter for spatial refinement
  • factor_tail = 1: Factor for tail estimator comparison
  • tail_extension = [10, 2]: Extension parameters for [0] mode and others
  • maxdofs = 1.0e4: Maximum degrees of freedom
  • nsamples = 150: Number of samples for error computation
  • use_equilibration_estimator = false: Use equilibration error estimator instead of standard residual estimator

Solver Options

  • use_iterative_solver = true: Use iterative solver (instead of a direct solver with full matrix)
  • bonus_quadorder_a = 2: Additional quadrature order for coefficient
  • bonus_quadorder_f = 0: Additional quadrature order for right-hand side

Right-Hand Side

  • f = (result, qpinfo) -> (result[1] = 1): Right-hand side function

Visualization

  • Plotter = nothing: Plotting backend (e.g., CairoMakie)

Results Storage

Results are automatically stored using DrWatson.jl in the data folder. The filename includes most problem parameters for reproducibility:

data/[problem]/[domain]/order=[order]_maxdofs=[maxdofs]_decay=[decay]_mean=[mean]_θ=([θ_spatial],[θ_stochastic])_tail=[tail_extension]

If the equilibration estimator is used, _eq is appended to the filename.