16 research outputs found
A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity
We study the iterative solution of linear systems of equations arising from
stochastic Galerkin finite element discretizations of saddle point problems. We
focus on the Stokes model with random data parametrized by uniformly
distributed random variables and discuss well-posedness of the variational
formulations. We introduce a Bramble-Pasciak conjugate gradient method as a
linear solver. It builds on a non-standard inner product associated with a
block triangular preconditioner. The block triangular structure enables more
sophisticated preconditioners than the block diagonal structure usually applied
in MINRES methods. We show how the existence requirements of a conjugate
gradient method can be met in our setting. We analyze the performance of the
solvers depending on relevant physical and numerical parameters by means of
eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of
the relevant preconditioned sub-matrices. We illustrate our findings using the
flow in a driven cavity as a numerical test case, where the viscosity is given
by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a
Bramble-Pasciak conjugate gradient method with block triangular preconditioner
outperforms a MINRES method with block diagonal preconditioner in terms of
iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
Convergence of adaptive stochastic Galerkin FEM
We propose and analyze novel adaptive algorithms for the numerical solution
of elliptic partial differential equations with parametric uncertainty. Four
different marking strategies are employed for refinement of stochastic Galerkin
finite element approximations. The algorithms are driven by the energy error
reduction estimates derived from two-level a posteriori error indicators for
spatial approximations and hierarchical a posteriori error indicators for
parametric approximations. The focus of this work is on the mathematical
foundation of the adaptive algorithms in the sense of rigorous convergence
analysis. In particular, we prove that the proposed algorithms drive the
underlying energy error estimates to zero
Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients
Solving high-dimensional random parametric PDEs poses a challenging
computational problem. It is well-known that numerical methods can greatly
benefit from adaptive refinement algorithms, in particular when functional
approximations in polynomials are computed as in stochastic Galerkin and
stochastic collocations methods. This work investigates a residual based
adaptive algorithm used to approximate the solution of the stationary diffusion
equation with lognormal coefficients. It is known that the refinement procedure
is reliable, but the theoretical convergence of the scheme for this class of
unbounded coefficients remains a challenging open question. This paper advances
the theoretical results by providing a quasi-error reduction results for the
adaptive solution of the lognormal stationary diffusion problem. A
computational example supports the theoretical statement
On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting
A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data
We propose and analyse a fully adaptive strategy for solving elliptic PDEs
with random data in this work. A hierarchical sequence of adaptive mesh
refinements for the spatial approximation is combined with adaptive anisotropic
sparse Smolyak grids in the stochastic space in such a way as to minimize the
computational cost. The novel aspect of our strategy is that the hierarchy of
spatial approximations is sample dependent so that the computational effort at
each collocation point can be optimised individually. We outline a rigorous
analysis for the convergence and computational complexity of the adaptive
multilevel algorithm and we provide optimal choices for error tolerances at
each level. Two numerical examples demonstrate the reliability of the error
control and the significant decrease in the complexity that arises when
compared to single level algorithms and multilevel algorithms that employ
adaptivity solely in the spatial discretisation or in the collocation
procedure.Comment: 26 pages, 7 figure
Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation
Partial differential equations (PDEs) with inputs that depend on infinitely
many parameters pose serious theoretical and computational challenges.
Sophisticated numerical algorithms that automatically determine which
parameters need to be activated in the approximation space in order to estimate
a quantity of interest to a prescribed error tolerance are needed. For elliptic
PDEs with parameter-dependent coefficients, stochastic Galerkin finite element
methods (SGFEMs) have been well studied. Under certain assumptions, it can be
shown that there exists a sequence of SGFEM approximation spaces for which the
energy norm of the error decays to zero at a rate that is independent of the
number of input parameters. However, it is not clear how to adaptively
construct these spaces in a practical and computationally efficient way. We
present a new adaptive SGFEM algorithm that tackles elliptic PDEs with
parameter-dependent coefficients quickly and efficiently. We consider
approximation spaces with a multilevel structure---where each solution mode is
associated with a finite element space on a potentially different mesh---and
use an implicit a posteriori error estimation strategy to steer the adaptive
enrichment of the space. At each step, the components of the error estimator
are used to assess the potential benefits of a variety of enrichment
strategies, including whether or not to activate more parameters. No marking or
tuning parameters are required. Numerical experiments for a selection of test
problems demonstrate that the new method performs optimally in that it
generates a sequence of approximations for which the estimated energy error
decays to zero at the same rate as the error for the underlying finite element
method applied to the associated parameter-free problem.Comment: 22 page