Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Cai, Zhiqiang; He, Cuiyu; Zhang, Shun

doi:10.1137/16m1056171

Cited by 45 publications

(27 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Next, we introduce the improved trace inequality. The proof of this lemma can be found in [11,Lemma 2.4] for the vector case; the proof of the tensor case is trival.…”

Section: 32mentioning

confidence: 99%

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

Gong

Mateos

et al. 2020

ESAIM: M2AN

View full text Add to dashboard Cite

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L 2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.AMS Subject Classification. 49J20 and 65N30.The dates will be set by the publisher.

show abstract

“…Next, we introduce the improved trace inequality. The proof of this lemma can be found in [11,Lemma 2.4] for the vector case; the proof of the tensor case is trival.…”

Section: 32mentioning

confidence: 99%

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

Gong

Mateos

et al. 2020

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…For the low regularity case, we introduced a speical projection operator in our earlier HDG work [18] to avoid the quantity q · n ∂T h in the analysis; however, this complicated the analysis. In this work, we use an improved inverse inequality from [5], and simplify the error analysis for the low regularity case. It is worth mentioning that part of our analysis (step 1 to step 3 in Section 3.3) improves the existing EDG error analysis by dealing with the case of low regularity solutions.…”

Section: Error Analysismentioning

confidence: 99%

“…However, the techniques in the previous EDG works are not applicable for the Dirichlet boundary control problem since the regularity of the solution may be low. Instead of introducing a special projection as in [18], we use an improved trace inequality from [5] to deal with the low regularity solution. We improve the existing EDG error analysis by dealing with the case of low regularity solutions; also this is the first work to give a rigorous error analysis for the IEDG method.…”

Section: Introductionmentioning

confidence: 99%

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

Chen

Singler

et al. 2019

J Sci Comput

View full text Add to dashboard Cite

We investigated an hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work [SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J. Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results. Recently, researchers have investigated discontinuous Galerkin (DG) methods for Dirichlet boundary control problems. We used a hybridizable discontinuous Galerkin (HDG) method for the Poisson equation in [25], and obtained a superlinear convergence rate for the control without using a special mesh or a higher order element. More recently, convection diffusion Dirichlet boundary control problems have gained more and more attention. Benner et al. in [3] used a local discontinuous Galerkin (LDG) method to obtain a sublinear convergence rate for the control. We considered an HDG method and proved optimal superlinear convergence rate for the control in [24] if the regularity of the solution is high, i.e., y ∈ H 1+s (Ω) with s ≥ 1/2. To overcome the difficulty for the low regularity case (0 ≤ s < 1/2), we utilized a special projection operator to get an optimal superlinear convergence rate in [18]; the numerical analysis was more complicated than in [24]. Furthermore, in contrast to [25], we obtained an optimal superliner convergence rate for the control by using a discontinuous higher order (quadratic) element.Although the degrees of freedom of the HDG method are significantly reduced compared to standard mixed methods, DG methods and LDG methods, they are still larger than the degrees of freedom of the CG method. In this work, we use embedded DG (EDG) and interior EDG (IEDG) methods to approximate the solution of the Dirichlet boundary control problem. The EDG and IEDG methods are obtained from the HDG methods, and the global systems both use the same continuous elements; this reduces the number of degrees of freedom considerably. To approximate the control, we use continuous element in the EDG method, and discontinuous elements in the IEDG method. Although the degrees of freedom of IEDG is slightly larger than the EDG method, the IEDG method provides greater flexibility for boundary control problems: we can use different finite element spaces for the control and the state. One possible benefit of the greater flexibility of the IEDG method is that discontinuous elements ...

show abstract

“…This low regularity may lead to reduced accuracy for numerical approximations [2,47]. In literature there are usually two types of methods to improve the numerical accuracy, interface(or body)-fitted methods [6,9,14,36,26,11] and interface-unfitted methods. For the interface-fitted methods, meshes aligned with the interface are used so as to dominate the approximation error caused by the non-smoothness of solution.…”

Section: Introductionmentioning

confidence: 99%

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

Wang

Yang

Xie

2019

Computational Methods in Applied Mathematics

View full text Add to dashboard Cite

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems, and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the L 2 norm are derived. Numerical results are provided to verify the theoretical results.

show abstract

Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

Cited by 45 publications

References 32 publications

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

Contact Info

Product

Resources

About