2,713 research outputs found
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
Stochastic Galerkin methods for non-affine coefficient representations are
known to cause major difficulties from theoretical and numerical points of
view. In this work, an adaptive Galerkin FE method for linear parametric PDEs
with lognormal coefficients discretized in Hermite chaos polynomials is
derived. It employs problem-adapted function spaces to ensure solvability of
the variational formulation. The inherently high computational complexity of
the parametric operator is made tractable by using hierarchical tensor
representations. For this, a new tensor train format of the lognormal
coefficient is derived and verified numerically. The central novelty is the
derivation of a reliable residual-based a posteriori error estimator. This can
be regarded as a unique feature of stochastic Galerkin methods. It allows for
an adaptive algorithm to steer the refinements of the physical mesh and the
anisotropic Wiener chaos polynomial degrees. For the evaluation of the error
estimator to become feasible, a numerically efficient tensor format
discretization is developed. Benchmark examples with unbounded lognormal
coefficient fields illustrate the performance of the proposed Galerkin
discretization and the fully adaptive algorithm
A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization
A deterministic method is proposed for solving the Boltzmann equation. The
method employs a Galerkin discretization of the velocity space and adopts, as
trial and test functions, the collocation basis functions based on weights and
roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or
full-range Hermite polynomials depending whether or not the distribution
function presents a discontinuity in the velocity space. The resulting
semi-discrete Boltzmann equation is in the form of a system of hyperbolic
partial differential equations whose solution can be obtained by standard
numerical approaches. The spectral rate of convergence of the results in the
velocity space is shown by solving the spatially uniform homogeneous relaxation
to equilibrium of Maxwell molecules. As an application, the two-dimensional
cavity flow of a gas composed by hard-sphere molecules is studied for different
Knudsen and Mach numbers. Although computationally demanding, the proposed
method turns out to be an effective tool for studying low-speed slightly
rarefied gas flows
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