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\n We consider a fully practical finite element approximation of the degenerate Cahn\u2013Hilliard equation with elasticity: Find the conserved order parameter,\n \n \n \n \n \n \u03b8\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n [<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\theta (x,t)\\in [-1,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and the displacement field,\n \n \n \n \n \n u<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n \\underline {u}(x,t) \\in \\mathbb {R}^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , such that\n \n \n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n \n \n \n \n \n \u2202\n \n <\/mml:mi>\n \n \u03b8\n \n <\/mml:mi>\n <\/mml:mrow>\n \n \n \u2202\n \n <\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:mstyle>\n <\/mml:mtd>\n \n a<\/mml:mi>\n m<\/mml:mi>\n p<\/mml:mi>\n ;<\/mml:mo>\n =<\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n \n .<\/mml:mo>\n \n (<\/mml:mo>\n b<\/mml:mi>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \n \u2207\n \n <\/mml:mi>\n \n [<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n \n \n \u0394\n \n <\/mml:mi>\n \n \u03b8\n \n <\/mml:mi>\n +<\/mml:mo>\n \n \n \u03b3\n \n <\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \n \n \u03a8\n \n <\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \n \n c<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \n E<\/mml:mi>\n <\/mml:mrow>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n )<\/mml:mo>\n :<\/mml:mo>\n \n \n \n E<\/mml:mi>\n <\/mml:mrow>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n )<\/mml:mo>\n ]<\/mml:mo>\n \n )<\/mml:mo>\n \n ,<\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n \n .<\/mml:mo>\n \n (<\/mml:mo>\n c<\/mml:mi>\n (<\/mml:mo>\n \n \u03b8\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \n \n \n \n E<\/mml:mi>\n <\/mml:mrow>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mstyle>\n <\/mml:mtd>\n \n a<\/mml:mi>\n m<\/mml:mi>\n p<\/mml:mi>\n ;<\/mml:mo>\n =<\/mml:mo>\n \n 0<\/mml:mn>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \n ,<\/mml:mo>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n \\begin{align*} \\gamma \\,\\textstyle {\\frac {\\partial \\theta }{\\partial t}} & = \\nabla \\,.\\, (b(\\theta ) \\,\\nabla \\,[- \\gamma \\, \\Delta \\theta + \\gamma ^{-1}\\, \\Psi \u2019(\\theta ) + \\textstyle \\frac 12\\,c\u2019(\\theta )\\,\\mathcal {C} \\,\\underline {\\underline {\\mathcal {E}}} (\\underline {u}): \\underline {\\underline {\\mathcal {E}}} (\\underline {u})]\\,) \\,, \\nonumber \\nabla \\,.\\, (c(\\theta ) \\,\\mathcal {C}\\, \\underline {\\underline {\\mathcal {E}}} (\\underline {u}) ) & = \\underline {0}\\,, \\end{align*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n subject to an initial condition\n \n \n \n \n \n \n \u03b8\n \n <\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \u22c5\n \n <\/mml:mo>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n [<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\theta ^0(\\cdot ) \\in [-1,1]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on\n \n \n \n \n \u03b8\n \n <\/mml:mi>\n \\theta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and boundary conditions on both equations. Here\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \\gamma \\in {\\mathbb R}_{>0}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the interfacial parameter,\n \n \n \n \n \u03a8\n \n <\/mml:mi>\n \\Psi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a non-smooth double well potential,\n \n \n \n \n \n \n E<\/mml:mi>\n <\/mml:mrow>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n \\underline {\\underline {\\mathcal {E}}}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the symmetric strain tensor,\n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the possibly anisotropic elasticity tensor,\n \n \n \n \n c<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n :=<\/mml:mo>\n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mstyle>\n <\/mml:mrow>\n c(s):=c_0+\\textstyle \\frac 12\\,(1-c_0)\\,(1+s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n \n c<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n \n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n c_0(\\gamma )\\in {\\mathbb R}_{>0}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n b<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n :=<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n s<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n b(s):=1-s^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.\n <\/p>","DOI":"10.1090\/s0025-5718-05-01802-8","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"7-41","source":"Crossref","is-referenced-by-count":16,"title":["Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid"],"prefix":"10.1090","volume":"75","author":[{"given":"John","family":"Barrett","sequence":"first","affiliation":[]},{"given":"Harald","family":"Garcke","sequence":"additional","affiliation":[]},{"given":"Robert","family":"N\u00fcrnberg","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,10,12]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1016\/0022-247X(77)90173-1","article-title":"Cone conditions and properties of Sobolev spaces","volume":"61","author":"Adams, R. 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