Hybridizable Discontinuous Galerkin Approximation for the Biharmonic Operator and Some Applications

Project: Research

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The biharmonic operator is an important model problem. It appears in von Kármán equations, Cahn–Hilliard equation, and the hydrodynamics coupled phase-field model, etc. So it is desirable to have efficient, robust numerical approximation to the biharmonic operator. Most existing numerical methods solving the biharmonic operator belong to the following categories: C1-conforming element methods, Ciarlet Raviart (C-R) methods, Hellan- Herrmann-Johnson (HHJ) methods, nonconforming methods and DG methods. They have the following shortcomings. C1-conforming element methods are not easy to implement. C-R methods don’t provide discreteH2-stability of the numerical solution which is essentially needed to handle the non-linearity of von Kármán equations, unless the domain is a convex polygon in plane. HHJ methods have too many global degrees of freedom since they approximate both solution and the Hessian of solution. Most nonconforming methods and DG methods approximate the solution only such that their approximations to the Laplacian of solution have much lower accuracy, while accurate approximation to the Lapcian of solution is desired for solving Cahn–Hilliard equation, the hydrodynamics coupled phase-field model.   The long-term goal of the P.I. is to develop and study, theoretically as well as computationally, efficient, robust numerical approximation of the biharmonic operator to avoid the above mentioned shortcomings of most existing numerical methods. In this proposal, the P.I. will concentrate on the study and development of new, emerging numerical scheme - hybridizable discontinuous Galerkin (HDG) methods. The HDG methods we propose for the biharmonic operator may avoid the above mentioned shortcomings. After hybridization, the global unknowns of our HDG methods for the biharmonic operator are numerical approximations to the traces of solution and its Laplacian along mesh interfaces, such that their global degrees of freedom are not large. Numerical approximations to both solution and its Laplacian are provided. With careful designed numerical solution spaces and numerical fluxes, our HDG methods for biharmonic operator will provide the standard discreteH2-stability of the numerical solution on arbitrary Lipschitz polyhedra and obtain optimal convergence to both the solution and its Laplacian. These features make our HDG methods suitable for numerically solving von Kármán equations, Cahn–Hilliard equation, the hydrodynamics coupled phase-field model and other elliptic or parabolic problems having biharmonic operator as their leading terms.   The P.I. is interested in studying and developing HDG methods, which are efficient, easy to implement and robust, for a variety of elliptic and parabolic equations, which get involved with biharmonic operator.                                2000 Mathematics Subject Classification. 65N30, 65L12.Key words and phrases. Hybridizable Discontinuous Galerkin method, biharmonic operator, discrete H2-stability, von Kármán equations, hydrodynamics coupled phase-field model. 


Project number9043181
Grant typeGRF
Effective start/end date1/01/22 → …