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In this paper, we develop an energy stable fully discrete discontinuous Galekin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.

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We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the KarhunenLoeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the