Hp-version Discontinuous Galerkin Finite Element Research Articles

Discontinuous Galerkin discretization in time of systems of second-order nonlinear hyperbolic equations

In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a hp-version discontinuous Galerkin finite element approximation in the time direction with an H1(Ω)-conforming finite element approximation in the spatial variables. Error bounds at the temporal nodal points are derived under a weak restriction on the temporal step size in terms of the spatial mesh size. Numerical experiments are presented to verify the theoretical results.

Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid hp-version discontinuous Galerkin finite element method from Congreve et al. [1] for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an hp-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed hp-adaptive two-grid method.

Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ . The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ ; thereby, only a linear system of equations is solved on the richer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ . In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ and $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ , respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.

Exponential convergence of hp-DGFEM for elliptic problems in polyhedral domains

We review the recent results of D. Schotzau et al. (hp-dGFEM for elliptic problems in polyhedra. I: Stability and quasioptimality on geometric meshes. Technical report 2009-28, Seminar for applied mathematics, ETH Zurich, 2009. To appear in SIAM J Numer Anal, 2013; hp-dGFEM for elliptic problems in polyhedra. II: Exponential convergence. Technical report 2009-29, Seminar for applied mathematics, ETH Zurich, 2009. To appear in SIAM J Numer Anal, 2013), and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and constant coefficients in three-dimsional and axiparallel polyhedra. The exponential rates are confirmed in a series of numerical tests.

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

We develop a one-parameter family of hp-version discontinuous Galerkin finite element methods, parameterised by θ ∈ [−1, 1], for the numerical solution of quasilinear elliptic equations in divergence form on a bounded open set Ω ⊂ ℝd, d ≥ 2. In particular, we consider the analysis of the family for the equation −∇ ·{μ(x, |∇u|)∇u} = f(x) subject to mixed Dirichlet–Neumann boundary conditions on ∂ Ω. It is assumed that μ is a real-valued function, μ ∈ C(Ω̄ × [0, ∞)), and there exist positive constants mμ and Mμ such that mμ(t − s) ≤ μ(x, t)t − μ(x, s)s ≤ Mμ(t − s) for t ≥ s ≥ 0 and all x ∈ Ω̄. Using a result from the theory of monotone operators for any value of θ ∈ [−1, 1], the corresponding method is shown to have a unique solution uDG in the finite element space. If u ∈ C1(Ω) ∩ Hk(Ω), k ≥ 2, then with discontinuous piecewise polynomials of degree p ≥ 1, the error between u and uDG, measured in the broken H1(Ω)-norm, is 𝒪(hs−1/pk−3/2), where 1 ≤ s ≤ min {p + 1, k}.

An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity

We develop the a posteriori error estimation of mixed hp-version discontinuous Galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders, and are independent of the locking parameter. A series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp-adaptive refinement procedure.

Hp -Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.