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In this paper, we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for convection-diffusion equations with inhomogeneous Dirichlet, Neumann and Robin boundary conditions, along with high-contrast coefficients. For time independent problems, boundary correctors $\mathcal{D}^m$ and $\mathcal{N}^{m}$ for Dirichlet, Neumann, and Robin conditions are designed. For time dependent problems, a scheme to update the boundary correctors is formulated. Error analysis in both cases is given to show the first-order convergence in energy norm with respect to the coarse mesh size $H$ and second-order convergence in $L^2-$norm, as verified by numerical examples, with which different finite difference schemes are compared for temporal discretization. Nonlinear problems are also demonstrated in combination with Strang splitting., Comment: 36 pages.11 figures. Submitted to Journal of Computational Mathematics

In this paper, we combine the multiscale flnite element method to propose an algorithm for solving the non-stationary Stokes-Darcy model, where the permeability coefflcient in the Darcy region exhibits multiscale characteristics. Our algorithm involves two steps: first, conducting the parallel computation of multiscale basis functions in the Darcy region. Second, based on these multiscale basis functions, we employ an implicitexplicit scheme to solve the Stokes-Darcy equations. One signiflcant feature of the algorithm is that it solves problems on relatively coarse grids, thus signiflcantly reducing computational costs. Moreover, under the same coarse grid size, it exhibits higher accuracy compared to standard flnite element method. Under the assumption that the permeability coefflcient is periodic and independent of time, this paper demonstrates the stability and convergence of the algorithm. Finally, the rationality and effectiveness of the algorithm are verifled through three numerical experiments, with experimental results consistent with theoretical analysis.

This paper explores the application of the multiscale finite element method (MsFEM) to address steady-state Stokes-Darcy problems with BJS interface conditions in highly heterogeneous porous media. We assume the existence of multiscale features in the Darcy region and propose an algorithm for the multiscale Stokes-Darcy model. During the offline phase, we employ MsFEM to construct permeability-dependent offline bases for efficient coarse-grid simulation, with this process conducted in parallel to enhance its efficiency. In the online phase, we use the Robin-Robin algorithm to derive the model's solution. Subsequently, we conduct error analysis based on $L^2$ and $H^1$ norms, assuming certain periodic coefficients in the Darcy region. To validate our approach, we present extensive numerical tests on highly heterogeneous media, illustrating the results of the error analysis.

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by \emph{a posteriori} error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical.

In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equation. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then constraint energy minimization performed in oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layer is chosen properly, we can prove that the convergence rate is proportional to the coarse mesh size. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performances of our method., Comment: arXiv admin note: text overlap with arXiv:1704.03193

In this paper, we study the generalized multiscale finite element method (GMsFEM) for single phase compressible flow in highly heterogeneous porous media. We follow the major steps of the GMsFEM to construct permeability dependent offline basis for fast coarse-grid simulation. The offline coarse space is efficiently constructed only once based on the initial permeability field with parallel computing. A rigorous convergence analysis is performed for two types of snapshot spaces. The analysis indicates that the convergence rates of the proposed multiscale method depend on the coarse meshsize and the eigenvalue decay of the local spectral problem. To further increase the accuracy of multiscale method, residual driven online multiscale basis is added to the offline space. The construction of online multiscale basis is based on a carefully design error indicator motivated by the analysis. We find that online basis is particularly important for the singular source. Rich numerical tests on typical 3D highly heterogeneous medias are presented to demonstrate the impressive computational advantages of the proposed multiscale method.

In this paper, a strongly mass conservative and stabilizer free scheme is designed and analyzed for the coupled Brinkman-Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered DG scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve a sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method.

In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method., Comment: 25 pages, 7 figures

In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier-Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied without any postprocessing. The resulting method is pressure-robust so that the pressure approximation does not influence the velocity approximation. A new nonlinear convective term that earning non-negativity is proposed. The optimal convergence estimates for all the variables in $L^2$ norm are proved. Also, assuming that the rotational part of the forcing term is small enough, we are able to prove that the velocity error is independent of the Reynolds number and of the pressure. Furthermore, superconvergence can be achieved for velocity under a suitable projection. Numerical experiments are provided to validate the theoretical findings and demonstrate the performances of the proposed method.

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realisation of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen-Lo\`eve expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.

In this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with stress-displacement-rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approximation developed in (L. Zhao and E.-J. Park, SIAM J. Sci. Comput. 42 (2020), A2158-A2181). The proposed method is locking free and can handle highly distorted grids possibly including hanging nodes, which is desirable for practical applications. We prove the convergence estimates for the semi-discrete scheme and fully discrete scheme for all the variables in their natural norms. In particular, the stability and convergence analysis do not need a uniformly positive storativity coefficient. Moreover, to reduce the size of the global system, we propose a five-field formulation based fixed stress splitting scheme, where the linear convergence of the scheme is proved. Several numerical experiments are carried out to confirm the optimal convergence rates and the locking-free property of the proposed method., Comment: 29 pages

In this paper, we develop the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to parabolic equations with heterogeneous diffusion coefficients. The construction of the method is based on two multiscale spaces: pressure multiscale space and velocity multiscale space. The pressure space is constructed via a set of well-designed local spectral problems, which can be solved independently. Based on the computed pressure multiscale space, we will construct the velocity multiscale space by applying constrained energy minimization. The convergence of the proposed method is proved.In particular, we prove that the convergence of the method depends only on the coarse grid size, and is independent of the heterogeneities and contrast of thediffusion coefficient. Four typical types of permeability fields are exploited in the numerical simulations, and the results indicate that our proposed method works well and gives efficient and accurate numerical solutions., Comment: 25 pages

In this paper we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confirm the theoretical findings., Comment: arXiv admin note: text overlap with arXiv:1911.08759

In this paper we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right hand side of the body force in the discrete formulation by exploiting divergence preserving velocity reconstruction operator, which is the crux for pressure independent velocity error estimates. The optimal convergence for velocity gradient, velocity and pressure are proved. In addition, we are able to prove the superconvergence of velocity approximation by the incorporation of divergence preserving velocity reconstruction operator in the dual problem, which is also an important contribution of this paper. Finally, several numerical experiments are carried out to confirm the theoretical findings., Comment: 23 pages

Modeling flows in fractured porous media is important in applications. One main challenge in numerical simulation is that the flow is strongly influenced by the fractures, so that the solutions typically contain complex features, which require high computational grid resolutions. Instead of using uniformly fine mesh, a more computationally efficient adaptively refined mesh is desirable. In this paper we design and analyze a novel residual-type a posteriori error estimator for staggered DG methods on general polygonal meshes for Darcy flows in fractured porous media. The method can handle fairly general meshes and hanging nodes can be simply incorporated into the construction of the method, which is highly appreciated for adaptive mesh refinement. The reliability and efficiency of the error estmator are proved. The derivation of the reliability hinges on the stability of the continuous setting in the primal formulation. A conforming counterpart that is continuous within each bulk domain for the discrete bulk pressure is defined to facilitate the derivation of the reliability. Finally, several numerical experiments including multiple non-intersecting fractures are carried out to confirm the proposed theories., Comment: 20 pages, 16 figures. arXiv admin note: text overlap with arXiv:2005.10955

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectively. Our current analysis weakens the usual assumption on the polygonal mesh, which can integrate more general meshes such as elements with arbitrarily small edges into our theoretical framework. We prove the optimal convergence estimates in $L^2$ error for all the variables by exploiting the Ritz projection. Importantly, our error estimates are shown to be fully robust with respect to the heterogeneity and anisotropy of the permeability coefficients. Several numerical experiments including meshes with small edges and anisotropic meshes are carried out to confirm the theoretical findings. Finally, our method is applied in the framework of unfitted mesh., Comment: 23 pages, 34 figures

In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general quadrilateral and polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in \cite{KimChung13}, we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal $L^2$ error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for $L^2$ error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented., Comment: 18 pages, 2 figures, 8 tables

Two residual-type error estimators for the mortar staggered discontinuous Galerkin discretizations of second order elliptic equations are developed. Both error estimators are proved to be reliable and efficient. Key to the derivation of the error estimator in potential $L^2$ error is the duality argument. On the other hand, an auxiliary function is defined, making it capable of decomposing the energy error into conforming part and nonconforming part, which can be combined with the well-known Scott-Zhang local quasi-interpolation operator and the mortar discrete formulation yields an error estimator in energy error. Importantly, our analysis for both error estimators does not require any saturation assumptions which are often needed in the literature. Several numerical experiments are presented to confirm our proposed theories., Comment: 17 pages, 15 figures

In this paper we develop a staggered discontinuous Galerkin method for the Stokes and Darcy-Forchheimer problems coupled with the \Red{Beavers-Joseph-Saffman} conditions. The method is defined by imposing staggered continuity for all the variables involved and the interface conditions are enforced by switching the roles of the variables met on the interface, which eliminate the hassle of introducing additional variables. This method can be flexibly applied to rough grids such as the highly distorted grids and the polygonal grids. In addition, the method allows nonmatching grids on the interface thanks to the special inclusion of the interface conditions, which is highly appreciated from a practical point of view. A new discrete trace inequality and a generalized Poincar\'{e}-Friedrichs inequality are proved, which enables us to prove the optimal convergence estimates under reasonable regularity assumptions. Finally, several numerical experiments are given to illustrate the performances of the proposed method, and the numerical results indicate that the proposed method is accurate and efficient, in addition, it is a good candidate for practical applications., Comment: 30 pages, 43 figures

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that $\kappa h$ is sufficiently small, where $\kappa$ is the wave number and $h$ is the mesh size. Error estimates for both the scalar and vector variables in $L^2$ norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions., Comment: 17 pages, 27 figures

In this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium. Our methodology is based on the recently developed non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. In our method, each coarse block is decomposed into various regions according to the contrast ratio, and we require that the contrast ratio should be relatively small within each region. The basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise. Numerical analysis shows that the resulting basis functions can be localizable and have a decay property. The convergence of the multiscale solution is also proved. Finally, some numerical experiments are carried out to illustrate the performances of the proposed method. They show that the proposed method can solve problem with high contrast medium efficiently. In particular, if the oversampling size is large enough, then we can achieve the desired error., Comment: 17 pages, 21 figures