A Unified Study of Continuous and Discontinuous Galerkin Methods

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.Comment: 39 page

A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problems

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of el- liptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the back- ground mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and con- dition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.Comment: 35 pages, 12 figures, 2 table

A stabilized P1P_1 immersed finite element method for the interface elasticity problems

We develop a new finite element method for solving planar elasticity problems involving of heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the `broken' Crouzeix-Raviart $P_1$-nonconforming finite element method for elliptic interface problems \cite{Kwak-We-Ch}. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method \cite{Arnold-IP},\cite{Ar-B-Co-Ma},\cite{Wheeler}. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace-Young condition along the interface of each element. We prove optimal $H^1$ and divergence norm error estimates. Numerical experiments are carried out to demonstrate that the our method is optimal for various Lam\`e parameters $\mu$ and $\lambda$ and locking free as $\lambda\to\infty$.Comment: Submitted to M2an on May 18 2015. Added a new author (Dae H. Kyeong

Analysis of a new stabilized discontinuous Galerkin method for the reaction-diffusion problem with discontinuous coefficient

In this paper, a new stabilized discontinuous Galerkin method within a new function space setting is introduced, which involves an extra stabilization term on the normal fluxes across the element interfaces. It is different from the general DG methods. The formulation satisfies a local conservation property and we prove well posedness of the new formulation by Inf-Sup condition. A priori error estimates are derived, which are verified by a 2D experiment on a reaction-diffusion type model problem

Discontinuous Galerkin methods for fractional elliptic problems

We provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under $L^{2}$ norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations

A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model

Modeling flow through porous media with multiple pore-networks has now become an active area of research due to recent technological endeavors like geological carbon sequestration and recovery of hydrocarbons from tight rock formations. Herein, we consider the double porosity/permeability (DPP) model, which describes the flow of a single-phase incompressible fluid through a porous medium exhibiting two dominant pore-networks with a possibility of mass transfer across them. We present a stable mixed discontinuous Galerkin (DG) formulation for the DPP model. The formulation enjoys several attractive features. These include: (i) Equal-order interpolation for all the field variables (which is computationally the most convenient) is stable under the proposed formulation. (ii) The stabilization terms are residual-based, and the stabilization parameters do not contain any mesh-dependent parameters. (iii) The formulation is theoretically shown to be consistent, stable, and hence convergent. (iv) The formulation supports non-conforming discretizations and distorted meshes. (v) The DG formulation has improved element-wise (local) mass balance compared to the corresponding continuous formulation. (vi) The proposed formulation can capture physical instabilities in coupled flow and transport problems under the DPP model.Comment: There was a mistake in the boundedness proof in the earlier version (specifically version #1). We now rectified this mistake and improved sections 1 and

A Cut Discontinuous Galerkin Method for the Laplace-Beltrami Operator

We develop a discontinuous cut finite element method (CutFEM) for the Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in $\mathbb{R}^d$. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighboring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.Comment: 25 pages, 3 figures, 3 table

Removing the stabilization parameter in fitted and unfitted symmetric Nitsche formulations

In many situations with finite element discretizations it is desirable or necessary to impose boundary or interface conditions not as essential conditions -- i.e. through the finite element space -- but through the variational formulation. One popular way to do this is Nitsche's method. In Nitsche's method a stabilization parameter $\lambda$ has to be chosen "sufficiently large" to provide a stable formulation. Sometimes discretizations based on a Nitsche formulation are criticized because of the need to manually choose this parameter. While in the discontinuous Galerkin community variants of the Nitsche method -- known as "interior penalty" method in the DG context -- are known which do not require such a manually chosen stabilization parameter, this has not been considered for Nitsche formulations in other contexts. We introduce and analyse such a parameter-free variant for two applications of Nitsche's method. First, the classical Nitsche formulation for the imposition of boundary conditions with fitted meshes and secondly, an unfitted finite element discretizations for the imposition of interface conditions is considered. The introduced variants of corresponding Nitsche formulations do not change the sparsity pattern and can easily be implemented into existing finite element codes. The benefit of the new formulations is the removal of the Nitsche stabilization parameter $\lambda$ while keeping the stability properties of the original formulations for a "sufficiently large" stabilization parameter $\lambda$.Comment: 12 pages, 1 figure, 1 table, submitted to ECCOMAS 2016 Proceeding

A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems

We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demon- strated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.Comment: 22 pages, 4 figures, 1 tabl

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The velocity solution and pressure solution on each side of the interface are separately expanded in the standard nonconforming piecewise linear polynomials and the piecewise constant polynomials, respectively. Harmonic weighted fluxes and arithmetic fluxes are used across the interface and cut edges (segment of the edges cut by the interface), respectively. Extra stabilization terms involving velocity and pressure are added to ensure the stable inf-sup condition. We show a priori error estimates under additional regularity hypothesis. Moreover, the errors {in energy and $L^2$ norms for velocity and the error in $L^2$ norm for pressure} are robust with respect to the viscosity {and independent of the location of the interface}. Results of numerical experiments are presented to {support} the theoretical analysis.Comment: 36 page