Galerkin Method For Elliptic Problems Research Articles

Hp-Version direct discontinuous Galerkin method for elliptic problems

hp-Version direct discontinuous Galerkin method for elliptic problems

A family of independent Variable Eddington Factor methods with efficient preconditioned iterative solvers

We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with the Discontinuous Galerkin transport discretization from [1] to form effective high-order, linear transport methods. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems presented by Arnold et al. [2] to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners [3], which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. We demonstrate that the VEF solution converges to the SN transport solution as the mesh is refined on both problems with smooth and non-smooth behavior in angle. Parallel performance studies show that the interior penalty VEF discretization's linear solve weak scales out to 1024 processors and strong scales well on a single node. Particular attention is paid to the parallel performance of the VEF algorithm when used in combination with a parallel block Jacobi transport sweep.

A staggered discontinuous Galerkin method for elliptic problems on rectangular grids

In this article, a staggered discontinuous Galerkin (SDG) approximation on rectangular meshes for elliptic problems in two dimensions is constructed and analyzed. The optimal convergence results with respect to discrete L2 and H1 norms are theoretically proved. Some numerical evidences to verify the optimal convergence rates are presented. Several numerical examples to the elliptic singularly perturbed problems with sharp boundary or interior layers are presented to show that the proposed SDG method is very effective, stable and accurate. Thanks to the simple structure of rectangular meshes, the discrete gradients across the boundaries of rectangular elements are easily defined, making numerical implementation much easier. The idea of using the rectangular meshes will be extended to more practical problems on a curved domain in future works.

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

In this paper, we introduce the multi-region discontinuous Galerkin composite finite element method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme.

Multiscale hybridizable discontinuous Galerkin method for elliptic problems in perforated domains

We propose a multiscale hybridizable discontinuous Galerkin method in perforated domains. Perforations may have various shapes and sizes. Following Generalized Multiscale Finite Element method, local snapshot spaces are constructed by solving local spectral problems. Then multiscale finite element spaces for numerical traces are built on the coarse edges within the framework of hybridizable discontinuous Galerkin method. Some representative examples are presented to investigate numerical performance of the proposed method.

Symmetric Interior Penalty Discontinuous Galerkin Methods for Elliptic Problems in Polygons

We analyze symmetric interior penalty discontinuous Galerkin (DG) finite element methods for linear, second-order elliptic boundary-value problems in polygons $\Omega$ with straight edges where solutions exhibit singular behavior near corners, and at boundary points where boundary conditions change. To resolve corner singularities, we admit both graded meshes and bisection refinement meshes. We prove that judiciously chosen refinement parameters in these mesh families imply optimal asymptotic rates of convergence with respect to the total number of degrees of freedom $N$, both for the DG energy norm error and the $L^2$-norm error. The sharpness of our asymptotic convergence rate estimates is confirmed in a series of numerical experiments.

A hybridizable direct discontinuous Galerkin method for elliptic problems

The aim of this work is to develop a hybridizable discontinuous Galerkin method for elliptic problems. In the proposed method, the numerical flux functions are constructed from the weak formulation of primal equation directly without converting the second-order equation to a first-order system. In order to guarantee the stability and convergence of the method, we derive a computable lower bound for the constant in numerical flux functions. We also establish a prior error estimation and give some theoretical analysis for the proposed method. Finally, a numerical experiment is presented to verify the theoretical results.

Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems

A polynomial preserving recovery technique is applied to an over-penalized symmetric interior penalty method. The discontinuous Galerkin solution values are used to recover the gradient and to further construct an a posteriori error estimator in the energy norm. In addition, for uniform triangular meshes and mildly structured meshes satisfying the $\epsilon$-$\sigma$ condition, the method for the linear element is superconvergent under the regular pattern and under the chevron pattern, while it is superconvergent for the quadratic element under the regular pattern.

High Order Discontinuous Galerkin Methods for Elliptic Problems on Surfaces

We derive and analyze high order discontinuous Galerkin methods for second order elliptic problems on implicitly defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of [D. N. Arnold et al., SIAM J. Numer. Anal., 39 (2002), pp. 1749--1779] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy and $L^2$ norms. Numerical results validating our theoretical estimates are also presented.

Couplings of mixed finite element and weak Galerkin methods for elliptic problems

In this paper, we consider two types of couplings for mixed finite element and weak Galerkin methods for elliptic problems. The first coupling simply connects the subdomain schemes on the interfaces by use of the numerical flux and pressure. The second one is based on a mortar space and imposes the weak continuity of the numerical flux. We derive solvability and a priori error estimates for both couplings.

Hp–Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains

In this article, we develop the a posteriori error estimation of hp–version discontinuous Galerkin composite finite element methods for the discretization of second‐order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. Although standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. Computable bounds on the error measured in terms of a natural (mesh‐dependent) energy norm are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp–adaptive refinement procedure will be presented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1342–1367, 2014

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized by small details in the computational domain or microstructures. In this setting, it is necessary to define a suitable coarse-level solver, in order to guarantee the scalability of the preconditioner under mesh refinement. To this end, we exploit recent ideas developed in the so-called composite finite element framework, which allows for the definition of finite element methods on general meshes consisting of agglomerated elements. Numerical experiments highlighting the practical performance of the proposed preconditioner are presented.

On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries. Rather than working with this three-field formulation, fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes, although other strategies could be devised. This procedure is applied to four elliptic problems, namely, the convection–diffusion equation (in the diffusion dominated regime), the Stokes problem, the Darcy problem and the Maxwell problem. We justify some well known dG methods with some modifications that in fact allow to improve the performance of the original methods, particularly when the physical properties are discontinuous.

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this paper we introduce the $hp$-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain $\Omega$ is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in $\Omega$. In this paper, we extend these ideas to the discontinuous Galerkin setting, based on employing the $hp$-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented.

Locally discontinuous but globally continuous Galerkin methods for elliptic problems

We propose and analyze a stabilized hybrid finite element method for elliptic problems consisting of locally discontinuous Galerkin problems in the primal variable coupled to a globally continuous problem in the multiplier. Numerical analysis shows that the proposed formulation preserves the main properties of the associate DG method such as consistency, stability, boundedness and optimal rates of convergence in the energy norm, and in the L2(Ω) norm for adjoint consistent formulations. For using an element based data structure, it has basically the same complexity and computational cost of classical conforming finite element methods. Convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here, with accuracy equivalent or even better than the corresponding DG approximations.

Pointwise a Posteriori Error Control for Discontinuous Galerkin Methods for Elliptic Problems

An a posteriori error bound for the maximum (pointwise) error for the interior penalty discontinuous Galerkin method for a standard elliptic model problem on polyhedral domains is presented. The computational domain is not required to be Lipschitz, thus allowing for domains with cracks and other irregular polyhedral domains. The proof is based on the direct use of Green's functions and varies substantially from the approach used in previous proofs of similar $L_\infty$ estimates for (continuous) finite element methods in the literature. Numerical experiments indicating the good behavior of the resulting a posteriori bounds within an adaptive algorithm are also presented.

A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1) $\mathcal{L}^{2}$ convergence rates for the solution and its gradient and O(h p+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems

We develop a smoothed aggregation-based algebraic multigrid solver for high-order discontinuous Galerkin discretizations of the Poisson problem. Algebraic multigrid is a popular and effective method for solving the sparse linear systems that arise from discretizing partial differential equations. However, high-order discontinuous Galerkin discretizations have proved challenging for algebraic multigrid. The increasing condition number of the matrix and loss of locality in the matrix stencil as p increases, in addition to the effect of weakly enforced Dirichlet boundary conditions all contribute to the challenging algebraic setting. We propose a smoothed aggregation approach that addresses these difficulties. In particular, the approach effectively coarsens degrees-of-freedom centered at the same spatial location as well as degrees-of-freedom at the domain boundary. Moreover, the character of the near null-space, particularly at the domain boundary, is captured by interpolation. One classic prolongation smoothing step of weighted-Jacobi is also shown to be ineffective at high-order, and a more robust energy-minimization approach is used, along with block relaxation that more directly utilizes the block diagonal structure of the discontinuous Galerkin discretization. Finally, we conclude by examining numerical results in support our proposed method.

Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems

Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems