C0 Interior Penalty Methods Research Articles

A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

C0 interior penalty methods for an elliptic distributed optimal control problem with general tracking and pointwise state constraints

We consider C0 interior penalty methods for a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of the domain. Error estimates and numerical results that illustrate the performance of the methods are presented.

Local parameter selection in the C0 interior penalty method for the biharmonic equation

Abstract The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages.

Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

A C0 interior penalty method for the phase field crystal equation

AbstractWe present a C0 interior penalty finite element method for the sixth‐order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature.

A C0 interior penalty method for mth-Laplace equation

In this paper, we propose a C0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝd, where m and d can be any positive integers. The standard H1-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing Dm of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete Hm-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete Hm-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.

A C0 interior penalty method for the Cahn-Hilliard equation

We present a C \begin{document}$ ^0 $\end{document} interior penalty method for the Cahn-Hilliard equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C \begin{document}$ ^0 $\end{document} interior penalty finite element method developed for the Cahn-Hilliard equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C \begin{document}$ ^0 $\end{document} interior penalty method. We furthermore support our conclusions with a few numerical experiments.

Adaptive [formula omitted] interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients

In this paper we conduct a priori and a posteriori error analysis of the C0 interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H2.

A cubic [formula omitted] interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints

We design and analyze a cubic C0 interior penalty method for linear–quadratic elliptic distributed optimal control problems with pointwise state and control constraints. Numerical results that corroborate the theoretical error estimates are also presented.

A QUADRATIC C0 INTERIOR PENALTY METHOD FOR THE QUAD-CURL PROBLEM

In this paper we study the C0 interior penalty method for a quad-curl problem arising from magnetohydrodynamics model on bounded polygons or polyhedrons. We prove the well-posedness of the numerical scheme and then derive the optimal error estimates in a discrete energy norm. A post-processing procedure that can produce C1 approximations is also presented. The performance of the method is illustrated by numerical experiments.

C0 interior penalty methods for an elliptic state-constrained optimal control problem with Neumann boundary condition

We study C0 interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains. The approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. We prove the convergence of numerical solutions with rates in the H2-like energy error by using the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm. Numerical experiments are provided to gauge the performance of the proposed methods.

Multigrid methods for a quad-curl problem based on [formula omitted] interior penalty method

In this paper we study the multigrid algorithm for a quad-curl problem based on C0 interior penalty method on bounded polygonal domains. It is shown that the contraction numbers of W-cycle multigrid algorithm for the C0 interior penalty method converge uniformly with rate of m−α, where m is the number of pre-smoothing (and post-smoothing) steps, α is the index of elliptic regularity. The performance of the multigrid fast solver is illustrated by numerical results.

A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.

A [formula omitted] interior penalty method for the Dirichlet control problem governed by biharmonic operator

An energy space based Dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a C0-interior penalty method is proposed and analyzed. An abstract a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular. Further an optimal order L2-norm error estimate is derived. Numerical experiments illustrate the theoretical findings.

A $$C^0$$ C 0 interior penalty method for a von Kármán plate

We investigate a quadratic $$C^0$$C0 interior penalty method for the approximation of isolated solutions of a von Karman plate. We prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates. Numerical results that illustrate the theoretical estimates are also presented.

A C0 interior penalty method for a fourth‐order variational inequality of the second kind

In this article, we propose a C0 interior penalty method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 36–59, 2016

Post-processing procedures for an elliptic distributed optimal control problem with pointwise state constraints

We consider an elliptic distributed optimal control problem with state constraints and compare three post-processing procedures that compute approximations of the optimal control from the approximation of the optimal state obtained by a quadratic C0 interior penalty method.

An interior penalty method for distributed optimal control problems governed by the biharmonic operator

In this paper, a C0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement.

Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction‐diffusion problems

Error estimates of finite element methods for reaction‐diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C0 interior penalty method with improved stability properties in comparison with the Galerkin FEM.