Finite element approximation of a phase field model arising in nanostructure patterning. Issue 6 (13th May 2015)

• Record Type: Journal Article
• Authors: Nürnberg, Robert; Tucker, Edward J. W.
• Is Part Of: Numerical methods for partial differential equations. Volume 31:Issue 6(2015)
• Page Start: 1890
• Page End: 1924
• Publication Date: 2015-05-13
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21972
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We consider a fully practical finite element approximation of the nonlinear parabolic Cahn–Hilliard system <disp-formula content-type="mathematics" id="num21972-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gn9f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>γ</mml:mo><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>−</mml:mo><mml:munder accentunder="true"><mml:mo>∇</mml:mo><mml:mo>_</mml:mo></mml:munder><mml:mo> </mml:mo><mml:mo>·</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:munder accentunder="true"><mml:mo>∇</mml:mo><mml:mo>_</mml:mo></mml:munder><mml:mo> </mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo> </mml:mo><mml:mo>, </mml:mo><mml:mo> </mml:mo><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo>γ</mml:mo><mml:mo>Δ</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mo>γ</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>Ψ</mml:mo><mml:mo>′</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mstyle scriptlevel="+1"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:mo>α</mml:mo><mml:mi>c</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>|</mml:mo><mml:munder accentunder="true"><mml:mo>∇</mml:mo><mml:mo>_</mml:mo></mml:munder><mml:mo> </mml:mo><mml:mo>ϕ</mml:mo><mml:msup><mml:mo>|</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo> </mml:mo><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula><disp-formula content-type="mathematics" id="num21972-disp-0002"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gng6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:munder accentunder="true"><mml:mo>∇</mml:mo><mml:mo>_</mml:mo></mml:munder><mml:mo> </mml:mo><mml:mo>·</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:munder accentunder="true"><mml:mo>∇</mml:mo><mml:mo>_</mml:mo></mml:munder><mml:mo> </mml:mo><mml:mo>ϕ</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo> </mml:mo><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula>subject to an initial condition <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gmzd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mo>.</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>, </mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> on the conserved order parameter <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gmvr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>, </mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, and mixed boundary conditions. Here, <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gmsn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>γ</mml:mo><mml:mo>∈</mml:mo><mml:msub><mml:mo>ℝ</mml:mo><mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> is the interfacial parameter, <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gmr3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>α</mml:mo><mml:mo>∈</mml:mo><mml:msub><mml:mo>ℝ</mml:mo><mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> is the field strength parameter, <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gn57" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>Ψ</mml:mo></mml:math></alternatives></inline-formula> is the obstacle potential, <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gn34" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> is the diffusion coefficient, and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gn11" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo>′</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>·</mml:mo><mml:mo>, </mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> denotes differentiation with respect to the second argument. Furthermore, <italic>w</italic> is the chemical potential and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gnwt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>ϕ</mml:mo></mml:math></alternatives></inline-formula> is the electrostatic potential. The system, in the context of nanostructure patterning, has been proposed to model the manipulation of morphologies in organic solar cells with the help of an applied electric field. In the limit <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2tk1gnxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21972:num21972-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>γ</mml:mo><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, it reduces to a sharp interface problem that models the evolution of an unstable interface between two dielectric media in the presence of a quasistatic electric field. On introducing a finite element approximation for the above Cahn–Hilliard system, we prove existence and stability of a discrete solution. Moreover, in the case of two space dimensions, we are able to prove convergence and hence existence of a solution to the considered system of partial differential equations. We demonstrate the practicality of our finite element approximation with several numerical simulations in two and three space dimensions. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1890–1924, 2015</p> </abstract>