Abstract: Quantifying uncertainty effects of coefficients that exhibit heterogeneity at multiple scales is among many outstanding challenges in subsurface flow models. Typically, the coefficients are modeled as functions of random variables governed by certain statistics. To quantify their uncertainty in the form of statistics (e.g., average fluid pressure or concentration) Monte-Carlo methods have been used. In a separate direction, multiscale numerical methods have been developed to efficiently capture spatial heterogeneity that otherwise would be intractable with standard numerical techniques. Since heterogeneity of individual realizations can differ drastically, a direct use of multiscale methods in Monte-Carlo simulations is problematic. Furthermore, Monte-Carlo methods are known to be very expensive as a lot of samples are required to adequately characterize the random component of the solution. In this study, we utilize a stochastic representation method that exploits the solution structure of the random process in order to construct a problem dependent stochastic basis. Using this stochastic basis representation a set of coupled yet deterministic equations is constructed. To reduce the computational cost of solving the coupled system, we develop a multiscale domain decomposition method utilizing Robin transmission conditions. In the proposed method, enrichment of the solution space can be performed at multiple levels that offer a balance between computational cost, and accuracy of the approximate solution.

Abstract: The standard nodal Lagrangian based continuous Galerkin finite element method (FEM) and control volume finite element method (CVFEM) are well known techniques for solving partial differential equations. Both of these methods have a common shortcoming in that the first derivative of the approximate solution of both methods is discontinuous. Further shortcomings of nodal Lagrangian bases arise when considering time dependent problems. For instance, increasing the degree of the basis in an effort to improve the accuracy of the approximate solution prohibits the use of common techniques such as mass matrix lumping. We introduce a $\mu^{\mathrm{th}}$ degree clamped basis-spline (B-spline) based analog of both the control volume finite element method and the continuous Galerkin finite element method in conjunction with a post processing technique which shall impose local conservation. The advantage of these techniques is that the B-spline basis is not only non-negative for any order $\mu$, and thus lends itself to mass matrix lumping for higher order basis functions, but also, for $\mu>2$, each basis function is smooth on the domain. We implement both the B-spline based CVFEM and FEM techniques as well as the post processing technique as they pertain to solving various two-point boundary value problems. A comparison of the convergence rates and properties of the error associated with satisfying local conservation is presented.

Abstract: Computational Fluid Dynamics (CFD) utilizes numerical solutions of Partial Differential Equations (PDE) on discretized volumes. These sets of discretized volumes, grids, can often contain tens of mil-lions, or billions of volumes. The analysis time of these large unstructured grids can take weeks to months to complete even on large computer clusters. For CFD solvers utilizing the Finite Volume Method (FVM) with implicit time stepping or a segregated pressure solver, a large portion of the computation time is spent solving a large linear system with a sparse coefficient matrix. In an effort to improve the performance of these CFD codes, in effect decreasing the time to solution of engineering problems, a conjugate gradient solver for a Finite Volume Method Solver Graphics Processing Units (GPU) was implemented to solve a model Poisson’s equation. Utilizing the improved memory throughput of NVIDIA’s Tesla K20 GPU a 2.5 times improvement was observed compared to a parallel CPU implementation on all 10 cores of an Intel Xeon E5-2670 v2. The parallel CPU implementation was constructed using the open source CFD toolbox, Open-FOAM.

Abstract: DarcyLite is a Matlab toolbox developed for numerical simulations of flow and transport in porous media in two dimensions. This paper focuses on the finite element methods and the corresponding code modules for solving the Darcy equation. Specifically, four major types of finite element solvers are presented: the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM). We further discuss the main design ideas and implementation strategies in DarcyLite. Numerical examples are included to demonstrate the usage and performance of this toolbox.

Abstract: In the present paper, a finite element model is developed based on a semi-discrete Streamline Upwind Petrov-Galerkin method to solve the fully-coupled two-dimensional shallow water and contaminant transport equations on a non-flat bed. The algorithm is applied on fixed computational meshes. Linear triangular elements are used to decompose the computational domain and a second-order backward differentiation implicit method is used for the time integration. The resulting nonlinear system is solved using a Newton-type method where the linear system is solved at each step using the Generalized Minimal Residual method. In order to examine the accuracy and robustness of the present scheme, numerical results are verified by different test cases.