Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

• Published in: Numerische Mathematik Vol. 135; no. 3; pp. 679 - 709
• Main Authors: Liu, Yuan, Chen, Wenbin, Wang, Cheng, Wise, Steven M.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017; Springer Nature B.V
• Subjects: Chemical potential; Convergence; Error analysis; Error functions; Finite element method; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Nonlinear analysis; Numerical Analysis; Numerical and Computational Physics; Simulation; Spatial resolution; Theoretical; 65K10; 65M60; 35K35; 65M12; 35K55
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-016-0813-2

We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343, 2012 ), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ℓ ∞ ( 0 , T ; L 2 ) ∩ ℓ 2 ( 0 , T ; H 2 ) error estimate, we perform a discrete ℓ ∞ ( 0 , T ; H 1 ) ∩ ℓ 2 ( 0 , T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h ) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo–Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian Δ h of the numerical solution, such that Δ h ϕ ∈ S h , for every ϕ ∈ S h , where S h is the finite element space.