12,983 research outputs found
Fast Online Generalized Multiscale Finite Element Method using Constraint Energy Minimization
Local multiscale methods often construct multiscale basis functions in the
offline stage without taking into account input parameters, such as source
terms, boundary conditions, and so on. These basis functions are then used in
the online stage with a specific input parameter to solve the global problem at
a reduced computational cost. Recently, online approaches have been introduced,
where multiscale basis functions are adaptively constructed in some regions to
reduce the error significantly. In multiscale methods, it is desired to have
only 1-2 iterations to reduce the error to a desired threshold. Using
Generalized Multiscale Finite Element Framework, it was shown that by choosing
sufficient number of offline basis functions, the error reduction can be made
independent of physical parameters, such as scales and contrast. In this paper,
our goal is to improve this. Using our recently proposed approach and special
online basis construction in oversampled regions, we show that the error
reduction can be made sufficiently large by appropriately selecting
oversampling regions. Our numerical results show that one can achieve a three
order of magnitude error reduction, which is better than our previous methods.
We also develop an adaptive algorithm and enrich in selected regions with large
residuals. In our adaptive method, we show that the convergence rate can be
determined by a user-defined parameter and we confirm this by numerical
simulations. The analysis of the method is presented
A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting
In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous
Galerkin (DG) method for multi-dimensional linear transport equations without
operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017)
to the 2D time dependent incompressible Euler equations in the vorticity-stream
function formulation and the guiding center Vlasov model. We adopt a local DG
method for Poisson's equation of these models. For tracing the characteristics,
we adopt a high order characteristics tracing mechanism based on a
prediction-correction technique. The SLDG with large time-stepping size might
be subject to extreme distortion of upstream cells. To avoid this problem, we
propose a novel adaptive time-stepping strategy by controlling the relative
deviation of areas of upstream cells.Comment: arXiv admin note: text overlap with arXiv:1709.0253
Diffuse interface models of locally inextensible vesicles in a viscous fluid
We present a new diffuse interface model for the dynamics of inextensible
vesicles in a viscous fluid. A new feature of this work is the implementation
of the local inextensibility condition in the diffuse interface context. Local
inextensibility is enforced by using a local Lagrange multiplier, which
provides the necessary tension force at the interface. To solve for the local
Lagrange multiplier, we introduce a new equation whose solution essentially
provides a harmonic extension of the local Lagrange multiplier off the
interface while maintaining the local inextensibility constraint near the
interface. To make the method more robust, we develop a local relaxation scheme
that dynamically corrects local stretching/compression errors thereby
preventing their accumulation. Asymptotic analysis is presented that shows that
our new system converges to a relaxed version of the inextensible sharp
interface model. This is also verified numerically. Although the model does not
depend on dimension, we present numerical simulations only in 2D. To solve the
2D equations numerically, we develop an efficient algorithm combining an
operator splitting approach with adaptive finite elements where the
Navier-Stokes equations are implicitly coupled to the diffuse interface
inextensibility equation. Numerical simulations of a single vesicle in a shear
flow at different Reynolds numbers demonstrate that errors in enforcing local
inextensibility may accumulate and lead to large differences in the dynamics in
the tumbling regime and differences in the inclination angle of vesicles in the
tank-treading regime. The local relaxation algorithm is shown to effectively
prevent this accumulation by driving the system back to its equilibrium state
when errors in local inextensibility arise.Comment: 25 page
Conservative Galerkin methods for dispersive Hamiltonian problems
An energy conservative discontinuous Galerkin scheme for a generalised third
order KdV type equation is designed. Based on the conservation principle, we
propose techniques that allow for the derivation of optimal a priori bounds for
the linear KdV equation and a posteriori bounds for the linear and modified KdV
equation. Extensive numerical experiments showcasing the good long time
behaviour of the scheme are summarised which are in agreement with the analysis
proposed
A posteriori modeling error estimates in the optimization of two-scale elastic composite materials
The a posteriori analysis of the discretization error and the modeling error
is studied for a compliance cost functional in the context of the optimization
of composite elastic materials and a two-scale linearized elasticity model. A
mechanically simple, parametrized microscopic supporting structure is chosen
and the parameters describing the structure are determined minimizing the
compliance objective. An a posteriori error estimate is derived which includes
the modeling error caused by the replacement of a nested laminate
microstructure by this considerably simpler microstructure. Indeed, nested
laminates are known to realize the minimal compliance and provide a benchmark
for the quality of the microstructures. To estimate the local difference in the
compliance functional the dual weighted residual approach is used. Different
numerical experiments show that the resulting adaptive scheme leads to simple
parametrized microscopic supporting structures that can compete with the
optimal nested laminate construction. The derived a posteriori error indicators
allow to verify that the suggested simplified microstructures achieve the
optimal value of the compliance up to a few percent. Furthermore, it is shown
how discretization error and modeling error can be balanced by choosing an
optimal level of grid refinement. Our two scale results with a single scale
microstructure can provide guidance towards the design of a producible
macroscopic fine scale pattern
Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations
This paper focuses on the adaptive discontinuous Galerkin (DG) methods for
the tempered fractional (convection) diffusion equations. The DG schemes with
interior penalty for the diffusion term and numerical flux for the convection
term are used to solve the equations, and the detailed stability and
convergence analyses are provided. Based on the derived posteriori error
estimates, the local error indicator is designed. The theoretical results and
the effectiveness of the adaptive DG methods are respectively verified and
displayed by the extensive numerical experiments. The strategy of designing
adaptive schemes presented in this paper works for the general PDEs with
fractional operators.Comment: 31 pages, 5 figure
Automated goal-oriented error control I: stationary variational problems
This article presents a general and novel approach to the automation of
goal-oriented error control in the solution of nonlinear stationary finite
element variational problems. The approach is based on automated linearization
to obtain the linearized dual problem, automated derivation and evaluation of a
posteriori error estimates, and automated adaptive mesh refinement to control
the error in a given goal functional to within a given tolerance. Numerical
examples representing a variety of different discretizations of linear and
nonlinear partial differential equations are presented, including Poisson's
equation, a mixed formulation of linear elasticity, and the incompressible
Navier-Stokes equations.Comment: 21 page
Application of the p-version of the finite-element method to global-local problems
A brief survey is given of some recent developments in finite-element analysis technology which bear upon the three main research areas under consideration in this workshop: (1) analysis methods; (2) software testing and quality assurance; and (3) parallel processing. The variational principle incorporated in a finite-element computer program, together with a particular set of input data, determines the exact solution corresponding to that input data. Most finite-element analysis computer programs are based on the principle of virtual work. In the following, researchers consider only programs based on the principle of virtual work and denote the exact displacement vector field corresponding to some specific set of input data by vector u(EX). The exact solution vector u(EX) is independent of the design of the mesh or the choice of elements. Except for very simple problems, or specially constructed test problems, vector u(EX) is not known. Researchers perform a finite-element analysis (or any other numerical analysis) because they wish to make conclusions concerning the response of a physical system to certain imposed conditions, as if vector u(EX) were known
Dynamic Implicit 3D Adaptive Mesh Refinement for Non-Equilibrium Radiation Diffusion
The time dependent non-equilibrium radiation diffusion equations are
important for solving the transport of energy through radiation in optically
thick regimes and find applications in several fields including astrophysics
and inertial confinement fusion. The associated initial boundary value problems
that are encountered often exhibit a wide range of scales in space and time and
are extremely challenging to solve. To efficiently and accurately simulate
these systems we describe our research on combining techniques that will also
find use more broadly for long term time integration of nonlinear multiphysics
systems: implicit time integration for efficient long term time integration of
stiff multiphysics systems, local control theory based step size control to
minimize the required global number of time steps while controlling accuracy,
dynamic 3D adaptive mesh refinement (AMR) to minimize memory and computational
costs, Jacobian Free Newton-Krylov methods on AMR grids for efficient nonlinear
solution, and optimal multilevel preconditioner components that provide level
independent solver convergence
Adaptive Finite Element Solution of Multiscale PDE-ODE Systems
We consider adaptive finite element methods for solving a multiscale system
consisting of a macroscale model comprising a system of reaction-diffusion
partial differential equations coupled to a microscale model comprising a
system of nonlinear ordinary differential equations. A motivating example is
modeling the electrical activity of the heart taking into account the chemistry
inside cells in the heart. Such multiscale models pose extremely
computationally challenging problems due to the multiple scales in time and
space that are involved.
We describe a mathematically consistent approach to couple the microscale and
macroscale models based on introducing an intermediate "coupling scale". Since
the ordinary differential equations are defined on a much finer spatial scale
than the finite element discretization for the partial differential equation,
we introduce a Monte Carlo approach to sampling the fine scale ordinary
differential equations. We derive goal-oriented a posteriori error estimates
for quantities of interest computed from the solution of the multiscale model
using adjoint problems and computable residuals. We distinguish the errors in
time and space for the partial differential equation and the ordinary
differential equations separately and include errors due to the transfer of the
solutions between the equations. The estimate also includes terms reflecting
the sampling of the microscale model. Based on the accurate error estimates, we
devise an adaptive solution method using a "blockwise" approach. The method and
estimates are illustrated using a realistic problem.Comment: 25 page
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