Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection–diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection–diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.     

An optimization-based numerical method for diffusion problems with sign-changing coefficients

A new optimization-based numerical method is proposed for the solution to diffusion problems with sign-changing conductivity coefficients. In contrast to existing approaches, our method does not rely on the discretization of a stabilized equation, and the convergence of the scheme can be proved without any symmetry assumption on the mesh near the interface where the conductivity sign changes.     

200 years of least squares method

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200 years of least squares method

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Solving Diffusion Problems on Rough Surfaces with a Hierarchical Multiscale FEM

Diffusion on rough surfaces is a basic problem for many applications in engineering and the sciences. Solving these problems with a standard finite element method is often difficult or even impossible, due to the computational work and the amount of memory needed to triangulate the whole surface with a mesh which resolves its oscillations. We discuss in this paper a hierarchical Finite Element Method of “heterogeneous multiscale” type, which only needs to resolve the surface's fine scale on small sampling domains within a macro triangulation of the underlying smooth surface. This method converges, for periodic surface roughness and sufficiently small amplitude, at a robust (i.e. scale independent) rate, to the homogenized solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L ² and the H ¹ norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.     

FE heterogeneous multiscale method for long-time wave propagation

A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our FE-HMM captures long-time dispersive effects of the true solution at a cost similar to that of a standard numerical homogenization scheme which, however, only captures the short-time macroscale behavior of the wave field.     

Numerical methods for stochastic partial differential equations with multiple scales

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.     

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L² norm. We then derive optimal a priori error estimates in the H¹ and L² norm for a FEM with variational crimes due to numerical integration. As an application, we derive a priori error estimates for a numerical homogenization method applied to nonlinear monotone elliptic problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 955–969, 2016     

Second order Chebyshev methods based on orthogonal polynomials

Summary. Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods. Received January 14, 2000 / Revised version received November 3, 2000 / Published online May 4, 2001