42 research outputs found
Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering
We present and analyze a pollution-free Petrov-Galerkin multiscale finite
element method for the Helmholtz problem with large wave number as a
variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous
finite elements at a coarse discretization scale as trial functions,
whereas the test functions are computed as the solutions of local problems at a
finer scale . The diameter of the support of the test functions behaves like
for some oversampling parameter . Provided is of the order of
and is sufficiently small, the resulting method is stable
and quasi-optimal in the regime where is proportional to . In
homogeneous (or more general periodic) media, the fine scale test functions
depend only on local mesh-configurations. Therefore, the seemingly high cost
for the computation of the test functions can be drastically reduced on
structured meshes. We present numerical experiments in two and three space
dimensions.Comment: The version coincides with v3. We only resized some figures which
were difficult to process for certain printer
The adaptive finite element method
Computer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other hand, a larger number of unknowns can improve the precision of the simulation. The adaptive finite element method (AFEM) is an algorithm for optimizing the choice of parameters so accurate simulation results can be obtained with as little computational effort as possible
Numerical approximation of planar oblique derivative problems in nondivergence form
A numerical method for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains is proposed. The numerical scheme employs a mixed formulation with piecewise affine functions on curved finite element domains. The direct approximation of the gradient of the solution turns the oblique derivative boundary condition into an oblique direction condition. A priori and a posteriori error estimates as well as numerical computations on uniform and adaptive meshes are provided
Numerical homogenization of H(curl)-problems
If an elliptic differential operator associated with an
-problem involves rough (rapidly varying)
coefficients, then solutions to the corresponding
-problem admit typically very low regularity, which
leads to arbitrarily bad convergence rates for conventional numerical schemes.
The goal of this paper is to show that the missing regularity can be
compensated through a corrector operator. More precisely, we consider the
lowest order N\'ed\'elec finite element space and show the existence of a
linear corrector operator with four central properties: it is computable,
-stable, quasi-local and allows for a correction of
coarse finite element functions so that first-order estimates (in terms of the
coarse mesh-size) in the norm are obtained provided
the right-hand side belongs to . With these four
properties, a practical application is to construct generalized finite element
spaces which can be straightforwardly used in a Galerkin method. In particular,
this characterizes a homogenized solution and a first order corrector,
including corresponding quantitative error estimates without the requirement of
scale separation
Numerical stochastic homogenization by quasilocal effective diffusion tensors
This paper proposes a numerical upscaling procedure for elliptic boundary
value problems with diffusion tensors that vary randomly on small scales. The
resulting effective deterministic model is given through a quasilocal discrete
integral operator, which can be further compressed to an effective partial
differential operator. Error estimates consisting of a priori and a posteriori
terms are provided that allow one to quantify the impact of uncertainty in the
diffusion coefficient on the expected effective response of the process
Localized implicit time stepping for the wave equation
This work proposes a discretization of the acoustic wave equation with
possibly oscillatory coefficients based on a superposition of discrete
solutions to spatially localized subproblems computed with an implicit time
discretization. Based on exponentially decaying entries of the global system
matrices and an appropriate partition of unity, it is proved that the
superposition of localized solutions is appropriately close to the solution of
the (global) implicit scheme. It is thereby justified that the localized (and
especially parallel) computation on multiple overlapping subdomains is
reasonable. Moreover, a re-start is introduced after a certain amount of time
steps to maintain a moderate overlap of the subdomains. Overall, the approach
may be understood as a domain decomposition strategy (in space and time) that
completely avoids inner iterations. Numerical examples are presented