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Discontinuous Galerkin finite element discretization of a degenerate Cahn-Hilliard equation with a single-well potential. (English) Zbl 1420.35429

Summary: This work concerns the analysis of a discontinuous Galerkin finite element approximation of a degenerate Cahn-Hilliard equation with single-well potential of the Lennard-Jones type. This equation is widely used for the diffuse interface modeling of solid tumors. A finite element discretization with discontinuous elements of the problem is developed, where the positivity of the solution, which is not straightforwardly guaranteed at the discrete level, is enforced through a variational inequality. The well posedness of the formulation is shown, together with the convergence to the weak solution. This discretization properly selects the solutions with a moving support with finite velocity, discarding the unphysical solutions with fixed support. The simulation results in two space dimensions are reported to test the validity of the proposed algorithm. Similar results as the ones obtained in standard phase ordering dynamics are found, highlighting the evolution of single domains to steady state with constant curvature. Imposing known solutions, good convergence properties of the discrete solution to the continuous one are observed using a proper norm calculated on its support. Contrarily to the discretizations with continuous elements, the use of discontinuous elements aims at recovering the optimal convergence rate found in literature for the finite element approximations of the Cahn-Hilliard equation with constant mobility. It is also useful when dealing with the dynamics of domains with corners, with highly heterogeneous materials and in presence of advective terms.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35K25 Higher-order parabolic equations
35K65 Degenerate parabolic equations
65K10 Numerical optimization and variational techniques
65G99 Error analysis and interval analysis
35D30 Weak solutions to PDEs
92C37 Cell biology
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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