40 research outputs found
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
Finite element convergence analysis for the thermoviscoelastic Joule heating problem
We consider a system of equations that model the temperature, electric
potential and deformation of a thermoviscoelastic body. A typical application
is a thermistor; an electrical component that can be used e.g. as a surge
protector, temperature sensor or for very precise positioning. We introduce a
full discretization based on standard finite elements in space and a
semi-implicit Euler-type method in time. For this method we prove optimal
convergence orders, i.e. second-order in space and first-order in time. The
theoretical results are verified by several numerical experiments in two and
three dimensions.Comment: 20 pages, 6 figures, 2 table
A generalized finite element method for linear thermoelasticity
We propose and analyze a generalized finite element method designed for
linear quasistatic thermoelastic systems with spatial multiscale coefficients.
The method is based on the local orthogonal decomposition technique introduced
by M{\aa}lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). We
prove convergence of optimal order, independent of the derivatives of the
coefficients, in the spatial -norm. The theoretical results are confirmed
by numerical examples
The Finite Element Method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation
We consider the time-dependent Gross-Pitaevskii equation describing the
dynamics of rotating Bose-Einstein condensates and its discretization with the
finite element method. We analyze a mass conserving Crank-Nicolson-type
discretization and prove corresponding a priori error estimates with respect to
the maximum norm in time and the - and energy-norm in space. The estimates
show that we obtain optimal convergence rates under the assumption of
additional regularity for the solution to the Gross-Pitaevskii equation. We
demonstrate the performance of the method in numerical experiments
Multiscale differential Riccati equations for linear quadratic regulator problems
We consider approximations to the solutions of differential Riccati equations
in the context of linear quadratic regulator problems, where the state equation
is governed by a multiscale operator. Similarly to elliptic and parabolic
problems, standard finite element discretizations perform poorly in this
setting unless the grid resolves the fine-scale features of the problem. This
results in unfeasible amounts of computation and high memory requirements. In
this paper, we demonstrate how the localized orthogonal decomposition method
may be used to acquire accurate results also for coarse discretizations, at the
low cost of solving a series of small, localized elliptic problems. We prove
second-order convergence (except for a logarithmic factor) in the
operator norm, and first-order convergence in the corresponding energy norm.
These results are both independent of the multiscale variations in the state
equation. In addition, we provide a detailed derivation of the fully discrete
matrix-valued equations, and show how they can be handled in a low-rank setting
for large-scale computations. In connection to this, we also show how to
efficiently compute the relevant operator-norm errors. Finally, our theoretical
results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs
from the previous one only by the addition of Remark 7.2 and minor changes in
formatting. 21 pages, 12 figure
Two-Level discretization techniques for ground state computations of Bose-Einstein condensates
This work presents a new methodology for computing ground states of
Bose-Einstein condensates based on finite element discretizations on two
different scales of numerical resolution. In a pre-processing step, a
low-dimensional (coarse) generalized finite element space is constructed. It is
based on a local orthogonal decomposition and exhibits high approximation
properties. The non-linear eigenvalue problem that characterizes the ground
state is solved by some suitable iterative solver exclusively in this
low-dimensional space, without loss of accuracy when compared with the solution
of the full fine scale problem. The pre-processing step is independent of the
types and numbers of bosons. A post-processing step further improves the
accuracy of the method. We present rigorous a priori error estimates that
predict convergence rates H^3 for the ground state eigenfunction and H^4 for
the corresponding eigenvalue without pre-asymptotic effects; H being the coarse
scale discretization parameter. Numerical experiments indicate that these high
rates may still be pessimistic.Comment: Accepted for publication in SIAM J. Numer. Anal., 201
Multiscale methods for problems with complex geometry
We propose a multiscale method for elliptic problems on complex domains, e.g.
domains with cracks or complicated boundary. For local singularities this paper
also offers a discrete alternative to enrichment techniques such as XFEM. We
construct corrected coarse test and trail spaces which takes the fine scale
features of the computational domain into account. The corrections only need to
be computed in regions surrounding fine scale geometric features. We achieve
linear convergence rate in energy norm for the multiscale solution. Moreover,
the conditioning of the resulting matrices is not affected by the way the
domain boundary cuts the coarse elements in the background mesh. The analytical
findings are verified in a series of numerical experiments
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