Scientific Computing II

Scientific Computing II

Summer term 2018 Priv.-Doz. Dr. Christian Rieger

Christopher Kacwin

Sheet 4

Exercise 1. (heterogeneous multiscale method)

We consider an open, and bounded domain Ω with triangulation T ∈ T_H and conti- nuous, piecewise linear finite elementsV_H(Ω) with zero boundary, as well asY = (0,1)ⁿ with triangulation K ∈ T_h and continuous, piecewise linear, periodic, zero-mean finite elements Wh(Y). Moreover, let A∈C⁰(Ω×Y)ⁿ be periodic in its second variable, uni- formly elliptic and A(x) =A(x, x/). We use the piecewise constant approximation on inner cells

A_h(x)|_x

T(K)=A(xT, x_T(yK)/) forT ∈ T_H and K ∈ T_h with corresponding barycenters xT,yK. uH ∈VH(Ω) is called an HMM-approximation if it solves

(f, vH)_L²_(Ω)=A_h(uH, vH) ∀v_H ∈VH(Ω), where

A_h(u_H, v_H) = X

T∈T_H

|T|

YT ,

A_h(x)∇_xR_T(u_H)(x)· ∇_xv_H(x) dx . (Here, R_T is the local reconstruction operator for δ =.)

Show that uH is the coarse part of the solution to the following two-scale problem:

Find (uH, u_h)∈VH(Ω)×VH(Ω, W_h(Y)) with ˆ

Ω

ˆ

Y

A_H(x, y)(∇_xu_H(x)+∇_yu_h(x, y))·(∇_xv_H(x)+∇_yv_h(x, y)) dydx= ˆ

Ω

f(x)v_H(x) dx for all (v_H, v_h)∈V_H(Ω)×V_H(Ω, W_h(Y)), withA_H(x, y)|_T×Y =A(x_T, y).

Show that one additionally has u_h(x, y)|_T×Y = 1

(R_T(u_H)−u_H)◦x_T(y−w₀) withw0 =xT/+ (1/2, . . . ,1/2)^>.

(16 points)

1