We analyze a fully discrete finite element numerical scheme for the Cahn-Hilliard-Stokes-Darcy system that models two-phase flows in coupled free flow and porous media. To avoid a well-known difficulty associated with the coupling between the Cahn-Hi
lliard equation and the fluid motion, we make use of the operator-splitting in the numerical scheme, so that these two solvers are decoupled, which in turn would greatly improve the computational efficiency. The unique solvability and the energy stability have been proved in~cite{CHW2017}. In this work, we carry out a detailed convergence analysis and error estimate for the fully discrete finite element scheme, so that the optimal rate convergence order is established in the energy norm, i.e.,, in the $ell^infty (0, T; H^1) cap ell^2 (0, T; H^2)$ norm for the phase variables, as well as in the $ell^infty (0, T; H^1) cap ell^2 (0, T; H^2)$ norm for the velocity variable. Such an energy norm error estimate leads to a cancellation of a nonlinear error term associated with the convection part, which turns out to be a key step to pass through the analysis. In addition, a discrete $ell^2 (0;T; H^3)$ bound of the numerical solution for the phase variables plays an important role in the error estimate, which is accomplished via a discrete version of Gagliardo-Nirenberg inequality in the finite element setting.
In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn-Hilliard-type equation with Flory-Huggins-deGennes energ
y potential. The numerical approach is based on a convex-concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.
We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differ
encing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficient linear numerical scheme with the help of an artificial regularization term of the form $Atau^kfrac{partial}{partial t}mathcal{L}^{p(k)}u$ where $mathcal{L}$ is the positive definite linear part of the flow, $tau$ is the uniform time step-size. The exponent $p(k)$ is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $mathcal{L}$ together with the desired temporal order of accuracy $k$. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results.
Multi-Grid Back-Projection (MGBP) is a fully-convolutional network architecture that can learn to restore images and videos with upscaling artifacts. Using the same strategy of multi-grid partial differential equation (PDE) solvers this multiscale ar
chitecture scales computational complexity efficiently with increasing output resolutions. The basic processing block is inspired in the iterative back-projection (IBP) algorithm and constitutes a type of cross-scale residual block with feedback from low resolution references. The architecture performs in par with state-of-the-arts alternatives for regression targets that aim to recover an exact copy of a high resolution image or video from which only a downscale image is known. A perceptual quality target aims to create more realistic outputs by introducing artificial changes that can be different from a high resolution original content as long as they are consistent with the low resolution input. For this target we propose a strategy using noise inputs in different resolution scales to control the amount of artificial details generated in the output. The noise input controls the amount of innovation that the network uses to create artificial realistic details. The effectiveness of this strategy is shown in benchmarks and it is explained as a particular strategy to traverse the perception-distortion plane.
Here, we describe our solution for the AIM-2019 Extreme Super-Resolution Challenge, where we won the 1st place in terms of perceptual quality (MOS) similar to the ground truth and achieved the 5th place in terms of high-fidelity (PSNR). To tackle thi
s challenge, we introduce the second generation of MultiGrid BackProjection networks (MGBPv2) whose major modifications make the system scalable and more general than its predecessor. It combines the scalability of the multigrid algorithm and the performance of iterative backprojections. In its original form, MGBP is limited to a small number of parameters due to a strongly recursive structure. In MGBPv2, we make full use of the multigrid recursion from the beginning of the network; we allow different parameters in every module of the network; we simplify the main modules; and finally, we allow adjustments of the number of network features based on the scale of operation. For inference tasks, we introduce an overlapping patch approach to further allow processing of very large images (e.g. 8K). Our training strategies make use of a multiscale loss, combining distortion and/or perception losses on the output as well as downscaled output images. The final system can balance between high quality and high performance.
We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equation
s that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $ell^2 (0,T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $ abla cdot (phi boldsymbol{u})$, we perform an $ell^infty (0,T; H_h^1)$ error estimate instead of the classical $ell^infty (0,T; ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.
We propose and analyze two novel decoupled numerical schemes for solving the Cahn-Hilliard-Stokes-Darcy (CHSD) model for two-phase flows in karstic geometry. In the first numerical scheme, we explore a fractional step method (operator splitting) to d
ecouple the phase-field (Cahn-Hilliard equation) from the velocity field (Stokes-Darcy fluid equations). To further decouple the Stokes-Darcy system, we introduce a first order pressure stabilization term in the Darcy solver in the second numerical scheme so that the Stokes system is decoupled from the Darcy system and hence the CHSD system can be solved in a fully decoupled manner. We show that both decoupled numerical schemes are uniquely solvable, energy stable, and mass conservative. Ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes.
We propose and study two second-order in time implicit-explicit (IMEX) methods for the coupled Stokes-Darcy system that governs flows in karst aquifers. The first is a combination of a second-order backward differentiation formula and the second-orde
r Gears extrapolation approach. The second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully-discretized schemes using finite element spatial discretizations are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes.
