Hp-version Interior Penalty Discontinuous Galerkin Research Articles

An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

Interior penalty discontinuous Galerkin FEMs for a gradient beam and CNTs

Carbon nanotubes (CNTs) are widely employed to modern engineering applications. The structural beam element is an acceptable model that has been used to simulate the CNTs' response. At the same time the gradient elasticity theory, taking into account the size effect phenomena, can be considered a viable option to study the mechanical response of CNTs through a generalized gradient beam model. Given that the constitutive equations of this theory produce a boundary layer, the hp-version interior penalty discontinuous Galerkin finite element methods (IPDGFEMs) are developed in this work for the solution of a static, tensile or compressive, gradient elastic beam in macro- and micro-structure (CNT), respectively. An a priori error analysis of the aforementioned methods is performed and numerical simulations that verify the analytical findings are carried out. The consistency, the stability and the convergence of the IPDGFEMs are proved. A priori error estimates established also indicate an optimal convergence in the meshsize h, yet a slightly suboptimal convergence in the polynomial degree p.

Hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes

We consider the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, new hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration. The proposed method employs elemental polynomial bases of total degree p (P_p-basis) defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. Numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a P_p-basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) Q_p-basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed hp-version DGFEM on general agglomerated meshes.

Hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analyzed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (𝒫p-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a 𝒫p-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

Discontinuous Galerkin methods on hp-anisotropic meshes II: a posteriori error analysis and adaptivity

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees. In particular, we exploit duality based hp-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment. The superiority of the proposed algorithm in comparison with standard hp-isotropic mesh refinement algorithms and an h-anisotropic/ p-isotropic adaptive procedure is illustrated by a series of numerical experiments.

A posteriori error analysis of hp-version discontinuous Galerkin finite-element methods for second-order quasi-linear elliptic PDEs

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

Hp-version interior penalty DGFEMs for the biharmonic equation

We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments.

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

In this article we consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with non-negative characteristic form under weak assumptions on the local mesh design and the local finite element spaces employed. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. To this end, we prove goal-oriented a priori hp-error bounds for linear target functionals of the solution on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a series of numerical experiments.

ENERGY NORM A POSTERIORI ERROR ESTIMATION OF hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS

In this paper, we develop the a posteriori error estimation of hp-version interior penalty discontinuous Galerkin discretizations of elliptic boundary-value problems. 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. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hp-adaptive refinement procedure.

A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second-order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors of arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalization parameter (that is present in the formulation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L 2 norm (the latter, for the symmetric version of the IP-DGFEM), provided that the solution belongs to an augmented Sobolev space.

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for second-order linear reaction-diffusion equations. To the best of our knowledge, the sharpest known error bounds for the hp-DGFEM are due to Riviere, Wheeler and Girault [9] and due to Houston, Schwab and Suli [6] which are optimal with respect to the meshsize h but suboptimal with respect to the polynomial degree p by half an order of p. We present improved error bounds in the energy norm, by introducing a new function space framework. More specifically, assuming that the solutions belong element-wise to an augmented Sobolev space, we deduce hp-optimal error bounds.