A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are L 2 stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal L 2 error estimate of O(h k+1 ) for polynomials of degree k for semi-discrete DG schemes, and the L 2 error of O(h k+1 + (∆t) 2 ) for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented.

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467-501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.1991 Mathematics Subject Classification. 65N12, 65N30, 35K35.

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ = (1), and a special initial guess for λ ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ = (1) and λ ≪ 1. The (m + 1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.

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In this paper, we study the biharmonic equation with the Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem into a system of Poisson equations. Different from the usual mixed method that leads to two Poisson problems but only applies to convex domains, the proposed decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A C 0 finite element algorithm is in turn proposed to solve the resulted system. In addition, we derive the optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.