Gem - International Journal on Geomathematics 

About: Gem - International Journal on Geomathematics is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Nonlinear system & Finite element method. It has an ISSN identifier of 1869-2672. Over the lifetime, 200 publications have been published receiving 2100 citations. The journal is also known as: International journal on geomathematics (Internet) & International journal on geomathematics (Print).

Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations

Abstract: The aim of this article is to give an overview of the current research towards applications of fractional partial differential equations for the modeling of anomalous heat transfer in porous media. We start with presenting a physical background behind the anomalous processes described by the Continuous Time Random Walk (CTRW) model and arguing its feasibility for modeling of heat transport processes in heterogeneous media. From the CTRW model on the microscopic level, a macroscopic model in form of a generalized fractional diffusion equation is then deduced. Both mathematical analysis of the generalized fractional diffusion equations and some methods for their numerical treatment are presented. Finally, some open questions and directions for further work are suggested.

Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method

Abstract: In this paper, we develop a multiscale finite element method for solving flows in fractured media. Our approach is based on generalized multiscale finite element method (GMsFEM), where we represent the fracture effects on a coarse grid via multiscale basis functions. These multiscale basis functions are constructed in the offline stage via local spectral problems following GMsFEM. To represent the fractures on the fine grid, we consider two approaches (1) discrete fracture model (DFM) (2) embedded fracture model (EFM) and their combination. In DFM, the fractures are resolved via the fine grid, while in EFM the fracture and the fine grid block interaction is represented as a source term. In the proposed multiscale method, additional multiscale basis functions are used to represent the long fractures, while short-size fractures are collectively represented by a single basis functions. The procedure is automatically done via local spectral problems. In this regard, our approach shares common concepts with several approaches proposed in the literature as we discuss. We would like to emphasize that our goal is not to compare DFM with EFM, but rather to develop GMsFEM framework which uses these (DFM or EFM) fine-grid discretization techniques. Numerical results are presented, where we demonstrate how one can adaptively add basis functions in the regions of interest based on error indicators. We also discuss the use of randomized snapshots (Calo et al. Randomized oversampling for generalized multiscale finite element methods, 2014), which reduces the offline computational cost.

Filtered hyperinterpolation: a constructive polynomial approximation on the sphere

Abstract: This paper considers a fully discrete filtered polynomial approximation on the unit sphere \({\mathbb{S}^{d}.}\) For \({f \in C(\mathbb{S}^{d}),V_{L,N}^{(a)} \, f}\) is a polynomial approximation which is exact for all spherical polynomials of degree at most L, so it inherits good convergence properties in the uniform norm for sufficiently smooth functions. The oscillations often associated with polynomial approximation of less smooth functions are localised by using a filter with support [0, a] for some a > 1, and with the value 1 on [0, 1]. The allowed choice of filters includes a recently introduced filter with minimal smoothness, and other smoother filters. The approximation uses a cubature rule with N points which is exact for all polynomials of degree \({t = \left\lceil{a L}\right\rceil+L-1.}\) The main theoretical result is that the uniform norm \({\|V_{L,N}^{(a)} \|}\) of the filtered hyperinterpolation operator is bounded independently of L, providing both good convergence and stability properties. Numerical experiments on \({\mathbb{S}^{2}}\) with a variety of filters, support intervals and cubature rules illustrate the uniform boundedness of the operator norm and the convergence of the filtered hyperinterpolation approximation for both an arbitrarily smooth function and a function with derivative discontinuities.

Generalized multiscale finite element method for elasticity equations

Abstract: In this paper, we discuss the application of generalized multiscale finite element method (GMsFEM) to elasticity equation in heterogeneous media. We consider steady state elasticity equations though some of our applications are motivated by elastic wave propagation in subsurface where the subsurface properties can be highly heterogeneous and have high contrast. We present the construction of main ingredients for GMsFEM such as the snapshot space and offline spaces. The latter is constructed using local spectral decomposition in the snapshot space. The spectral decomposition is based on the analysis which is provided in the paper. We consider both continuous Galerkin and discontinuous Galerkin coupling of basis functions. Both approaches have their cons and pros. Continuous Galerkin methods allow avoiding penalty parameters though they involve partition of unity functions which can alter the properties of multiscale basis functions. On the other hand, discontinuous Galerkin techniques allow gluing multiscale basis functions without any modifications. Because basis functions are constructed independently from each other, this approach provides an advantage. We discuss the use of oversampling techniques that use snapshots in larger regions to construct the offline space. We provide numerical results to show that one can accurately approximate the solution using reduced number of degrees of freedom.

A generalized multiscale finite element method for elastic wave propagation in fractured media

Abstract: In this paper, we consider elastic wave propagation in fractured media applying a linear-slip model to represent the effects of fractures on the wavefield. Fractured media, typically, are highly heterogeneous due to multiple length scales. Direct numerical simulations for wave propagation in highly heterogeneous fractured media can be computationally expensive and require some type of model reduction. We develop a multiscale model reduction technique that captures the complex nature of the media (heterogeneities and fractures) in the coarse scale system. The proposed method is based on the generalized multiscale finite element method, where the multiscale basis functions are constructed to capture the fine-scale information of the heterogeneous, fractured media and effectively reduce the degrees of freedom. These multiscale basis functions are coupled via the interior penalty discontinuous Galerkin method, which provides a block-diagonal mass matrix. The latter is needed for fast computation in an explicit time discretization, which is used in our simulations. Numerical results are presented to show the performance of the presented multiscale method for fractured media. We consider several cases where fractured media contain fractures of multiple lengths. Our numerical results show that the proposed reduced-order models can provide accurate approximations for the fine-scale solution.