Publication

MATHICSE Technical Report : Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

2014
Working Papers

Abstract

We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the solution at the scale of interest at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Optimal a priori error estimates in the L^2(H^1) and C^0(L^2) norm are derived taking into account the error due to time discretization as well as macro and micro spatial discretizations. Further, we present numerical simulations to illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

Official source

https://infoscience.epfl.ch/entities/publication/ff882f19-76e3-4d4e-a191-64efa362717e

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2014
Working Papers

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/ff882f19-76e3-4d4e-a191-64efa362717e

About this result

Related concepts (32)

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Discretization

In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).

Related publications (1)

MATHICSE Technical Report : Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

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MATHICSE Technical Report : Semi-implicit BDF time discretization of the Navier-Stokes equations with VMS-LES modeling in a High Performance Computing framework