793 research outputs found
Convergence of a discontinuous Galerkin multiscale method
A convergence result for a discontinuous Galerkin multiscale method for a
second order elliptic problem is presented. We consider a heterogeneous and
highly varying diffusion coefficient in with uniform spectral bounds and without any assumption on scale
separation or periodicity. The multiscale method uses a corrected basis that is
computed on patches/subdomains. The error, due to truncation of corrected
basis, decreases exponentially with the size of the patches. Hence, to achieve
an algebraic convergence rate of the multiscale solution on a uniform mesh with
mesh size to a reference solution, it is sufficient to choose the patch
sizes as . We also discuss a way to further
localize the corrected basis to element-wise support leading to a slight
increase of the dimension of the space. Improved convergence rate can be
achieved depending on the piecewise regularity of the forcing function. Linear
convergence in energy norm and quadratic convergence in -norm is obtained
independently of the forcing function. A series of numerical experiments
confirms the theoretical rates of convergence
A posteriori error control for fully discrete Crank–Nicolson schemes
We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank--Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, which is introduced in the present paper
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
Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems
The central objective of this paper is to develop reduced basis methods for
parameter dependent transport dominated problems that are rigorously proven to
exhibit rate-optimal performance when compared with the Kolmogorov -widths
of the solution sets. The central ingredient is the construction of
computationally feasible "tight" surrogates which in turn are based on deriving
a suitable well-conditioned variational formulation for the parameter dependent
problem. The theoretical results are illustrated by numerical experiments for
convection-diffusion and pure transport equations. In particular, the latter
example sheds some light on the smoothness of the dependence of the solutions
on the parameters
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