Analytical Approximation for the Generalized Laplace Equation with Step Function Coefficient

by: RF Sviercoski, CL Winter, AW Warrick

SIAM Journal on Applied Mathematics, Vol. 68, No. 5. (2008), pp. 1268-1281.

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Abstract

Many problems in science and engineering require the solution of the steady-state diffusion equation with a highly oscillatory coefficient. In this paper, we propose an analytical approximation $u(x) ∈ L^p(Ω)$, $1 ≤ p ≤ ∞$, for the generalized Laplace equation $∇ ⋅ (K\,(x)\,∇ u(x)) =0$ in $Ω ⊂ R^n$, with prescribed boundary conditions and the coefficient function $K(x) ∈ L^p(Ω)$ defined as a step function, not necessarily periodic. The proposed solution can be regarded as an approximation to the weak solution belonging to $W^1,p(Ω)$, the Sobolev space. When the coefficient function describes inclusions in a main matrix, then $K(x)$ is a periodic function, and such formulation leads to an approximation, in $L^p(Ω)$, to the solution of the periodic cell-problem, $∇ ⋅ (K(ε^-1x)∇\mathbfw(ε^-1x)) = ∇ ⋅ (K(ε^-1x)\mathbf1)$. The solution to the cell-problem is the key information needed to obtain the upscaled coefficient and therefore the zeroth-order approximation for a generalized elliptic equation with highly oscillating coefficient in $Ω ⊂ R^n$. Our numerical computation of the error between the proposed analytical approximation for the cell-problem, $w(ε^-1x)$ in $L^p(Ω)$, and the solution $w(ε^-1x)$ in $W^1,2(Ω)$, demonstrates to converge in the $L^2$-norm, when the scale parameter $ε$ approaches zero. The proposed approximation leads to the lower bound of the generalized Voigt–Reiss inequality, which is a more accurate two-sided estimate than the classical Voigt–Reiss inequality. As an application, we compute our approximate value for the homogenized coefficient when the heterogeneous coefficients are inclusions such as squares, circles, and lozenges, and we demonstrate that the results underestimate the effective coefficient with an error of 10% on average, when compared with published numerical results.

@article{citeulike:2797512, abstract = {Many problems in science and engineering require the solution of the steady-state diffusion equation with a highly oscillatory coefficient. In this paper, we propose an analytical approximation \$\tilde{u}(x) \in L^p(\Omega)\$, \$1 \leq p \leq \infty\$, for the generalized Laplace equation \$\nabla \cdot (K\,(x)\,\nabla u(x)) =0\$ in \$\Omega \subset R^n\$, with prescribed boundary conditions and the coefficient function \$K(x) \in L^p(\Omega)\$ defined as a step function, not necessarily periodic. The proposed solution can be regarded as an approximation to the weak solution belonging to \$W^{1,p}(\Omega)\$, the Sobolev space. When the coefficient function describes inclusions in a main matrix, then \$K(x)\$ is a periodic function, and such formulation leads to an approximation, in \$L^p(\Omega)\$, to the solution of the periodic cell-problem, \$\nabla \cdot (K(\varepsilon^{-1}x)\nabla\mathbf{w}(\varepsilon^{-1}x)) = \nabla \cdot (K(\varepsilon^{-1}x)\mathbf{1})\$. The solution to the cell-problem is the key information needed to obtain the upscaled coefficient and therefore the zeroth-order approximation for a generalized elliptic equation with highly oscillating coefficient in \$\Omega \subset R^n\$. Our numerical computation of the error between the proposed analytical approximation for the cell-problem, \$\tilde{w}(\varepsilon^{-1}x)\$ in \$L^{p}(\Omega)\$, and the solution \$w(\varepsilon^{-1}x)\$ in \$W^{1,2}(\Omega)\$, demonstrates to converge in the \$L^2\$-norm, when the scale parameter \$\varepsilon\$ approaches zero. The proposed approximation leads to the lower bound of the generalized Voigt\–Reiss inequality, which is a more accurate two-sided estimate than the classical Voigt\–Reiss inequality. As an application, we compute our approximate value for the homogenized coefficient when the heterogeneous coefficients are inclusions such as squares, circles, and lozenges, and we demonstrate that the results underestimate the effective coefficient with an error of 10\% on average, when compared with published numerical results.}, author = {Sviercoski, R. F. and Winter, C. L. and Warrick, A. W. }, citeulike-article-id = {2797512}, journal = {SIAM Journal on Applied Mathematics}, keywords = {electrostatics, mathematics}, number = {5}, pages = {1268--1281}, posted-at = {2008-05-14 12:00:34}, priority = {0}, publisher = {SIAM}, title = {Analytical Approximation for the Generalized Laplace Equation with Step Function Coefficient}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal\&id=SMJMAP000068000005001268000001\&idtype=cvips\&gifs=yes}, volume = {68}, year = {2008} }

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TY - JOUR ID - citeulike:2797512 L3 - citeulike-article-id:2797512 TI - Analytical Approximation for the Generalized Laplace Equation with Step Function Coefficient JF - SIAM Journal on Applied Mathematics VL - 68 IS - 5 SP - 1268 EP - 1281 PB - SIAM N2 - Many problems in science and engineering require the solution of the steady-state diffusion equation with a highly oscillatory coefficient. In this paper, we propose an analytical approximation $\tilde{u}(x) \in L^p(\Omega)$, $1 \leq p \leq \infty$, for the generalized Laplace equation $\nabla \cdot (K\,(x)\,\nabla u(x)) =0$ in $\Omega \subset R^n$, with prescribed boundary conditions and the coefficient function $K(x) \in L^p(\Omega)$ defined as a step function, not necessarily periodic. The proposed solution can be regarded as an approximation to the weak solution belonging to $W^{1,p}(\Omega)$, the Sobolev space. When the coefficient function describes inclusions in a main matrix, then $K(x)$ is a periodic function, and such formulation leads to an approximation, in $L^p(\Omega)$, to the solution of the periodic cell-problem, $\nabla \cdot (K(\varepsilon^{-1}x)\nabla\mathbf{w}(\varepsilon^{-1}x)) = \nabla \cdot (K(\varepsilon^{-1}x)\mathbf{1})$. The solution to the cell-problem is the key information needed to obtain the upscaled coefficient and therefore the zeroth-order approximation for a generalized elliptic equation with highly oscillating coefficient in $\Omega \subset R^n$. Our numerical computation of the error between the proposed analytical approximation for the cell-problem, $\tilde{w}(\varepsilon^{-1}x)$ in $L^{p}(\Omega)$, and the solution $w(\varepsilon^{-1}x)$ in $W^{1,2}(\Omega)$, demonstrates to converge in the $L^2$-norm, when the scale parameter $\varepsilon$ approaches zero. The proposed approximation leads to the lower bound of the generalized Voigt–Reiss inequality, which is a more accurate two-sided estimate than the classical Voigt–Reiss inequality. As an application, we compute our approximate value for the homogenized coefficient when the heterogeneous coefficients are inclusions such as squares, circles, and lozenges, and we demonstrate that the results underestimate the effective coefficient with an error of 10% on average, when compared with published numerical results. KW - electrostatics KW - mathematics AU - Sviercoski, RF AU - Winter, CL AU - Warrick, AW PY - 2008/// UR - http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SMJMAP000068000005001268000001&idtype=cvips&gifs=yes ER -

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