Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Apel, Thomas; Nicaise, Serge; Pfefferer, Johannes

doi:10.1002/num.22057

Cited by 21 publications

(32 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notice as well that (3.1) is not a conforming discretization of the very weak formulation of the state equation. However, according to [3,8], its applicability is guaranteed. In our discretized optimal control problem we aim to minimize the objective function…”

Section: A General Discretization Error Estimatementioning

confidence: 99%

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Apel¹,

Mateos²,

Pfefferer³

et al. 2018

Mathematical Control &Amp; Related Fields

Self Cite

View full text Add to dashboard Cite

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasiuniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided. of the adjoint state has a pole there, we getū ∈ H s (Γ) for all s < 3 2 . This regularity is determined by the larges convex angle and by the kinks due to the constraints.Unfortunately, this is not the whole truth. In exceptional cases, e. g. when the data enjoy certain symmetry, the leading singularity of type r 2/3 may not appear in the adjoint state. Instead, the solution may have a r 4/3 -singularity whose normal derivative has a r 1/3 -singularity which is not flattened by the projection Proj [a,b] . The control is less regular,ū ∈ H s (Γ) for all s < 5 6 . See Example 3.6 in [1]. Hence, dealing with these exceptional cases is not fun but necessary. If in the unconstrained case a stress intensity factor vanishes , i.e., the leading singularity does not occur, then the convergence result is still true, one may only see a better convergence in numerical tests. See Figure 3, right hand side and Remark 4.8. However, in the constrained case, the situation is the opposite. The exceptional case leads to the worstcase estimate. To deal with the "worst-case" and the "usual-case" in an unified way, we introduce in (2.4) some numbers related to the singular exponents.We distinguish two cases for the investigation of the discretization errors. After proving a general result in Section 3 we study the unconstrained case in Section 4 and the constrained case in Section 5. We focus on quasi-uniform meshes and distinguish general meshes and certain superconvergence meshes. In order not to overload the present paper, we postpone the study of graded meshes to [2]. The numerical tests in Section 6 confirm the theoretical results.The study of error estimates for Dirichlet control problems posed on polygonal domains can be traced back to [9], where a control constrained problem governed by a semilinear elliptic equation posed in a convex polygonal domain is studied. An order of convergence of h s is proved for all s < min(1, π/(2ω 1 )), where ω 1 is the largest interior angle, in both the control and the state variable. Later, in [18], it is proven that for unconstrained linear problems posed on convex domains, the state variable exhibits a better convergence property. The corresponding proof is based on a duality argument and estimates for the controls in weaker norms than L 2 (Γ). However, to the best of our knowledge, the argumentation is restricted to unconstrained problems. For the error of t...

show abstract

Section: A General Discretization Error Estimatementioning

confidence: 99%

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Apel¹,

Mateos²,

Pfefferer³

et al. 2018

Mathematical Control &Amp; Related Fields

Self Cite

View full text Add to dashboard Cite

show abstract

“…Numerical experiments confirm the theoretical results.The finite element solution y h is now searched in Y * h := Y h * ∩ Y h and is defined in the classical way: find(1.5)The same discretization was derived previously by Berggren [5] from a different point of view. In [2] we showed that the discretization error estimateholds for s = 1/2 if the domain is convex; this is a slight improvement of the result of Berggren, and the convex case is completely treated. In the case of non-convex domains this convergence order is reduced although the very weak solution y is also in H 1/2 (Ω); the finite element method does not lead to the best approximation in L 2 (Ω).…”

mentioning

confidence: 68%

“…with (w, v) G := G wv denoting the L 2 (G) scalar product or an appropriate duality product. In our previous paper [2] we showed that the appropriate space V for the test functions is…”

Section: Introductionmentioning

confidence: 99%

“…This sounds particularly simple in the case of mesh grading. However, the convergence proof in [2] contains not only interpolation error estimates for the dual solution and its normal derivative (which are improved now) but also the application of an inverse inequality which gives a too pessimistic result if used unchanged in the case of graded meshes. We prove in Section 2 a sharp result by using a weighted norm in intermediate steps.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Adapted Numerical Methods for the Poisson Equation with $L^2$ Boundary Data in NonConvex Domains

Apel¹,

Nicaise²,

Pfefferer³

2017

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

The very weak solution of the Poisson equation with L 2 boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the L 2 (Ω)-norm with order 1/2 in convex domains but has a reduced convergence order in non-convex domains although the solution remains to be contained in H 1/2 (Ω). The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases also with non-convex domains. Numerical experiments confirm the theoretical results.The finite element solution y h is now searched in Y * h := Y h * ∩ Y h and is defined in the classical way: find(1.5)The same discretization was derived previously by Berggren [5] from a different point of view. In [2] we showed that the discretization error estimateholds for s = 1/2 if the domain is convex; this is a slight improvement of the result of Berggren, and the convex case is completely treated. In the case of non-convex domains this convergence order is reduced although the very weak solution y is also in H 1/2 (Ω); the finite element method does not lead to the best approximation in L 2 (Ω). In order to describe the result we assume for simplicity that Ω has only one corner with interior angle ω ∈ (π, 2π). We proved in [2] the convergence order s = λ − 1/2 − ε, where

show abstract

“…The precise conditions on Ω, J and the Banach spaces involved will be given in the subsequent sections. Notice that both the exterior value problems (1.3b) and (1.4b [14,15,17] for the case s = 1). To the best of our knowledge this is the first work that considers the notion of very-weak solutions for nonlocal (fractional) exterior value problems associated with the fractional Laplace operator.…”

Section: Introductionmentioning

confidence: 99%

External optimal control of nonlocal PDEs

Antil¹,

Khatri²,

Warma³

2019

Inverse Problems

View full text Add to dashboard Cite

Very recently Warma [42] has shown that for nonlocal PDEs associated with the fractional Laplacian, the classical notion of controllability from the boundary does not make sense and therefore it must be replaced by a control that is localized outside the open set where the PDE is solved. Having learned from the above mentioned result, in this paper we introduce a new class of source identification and optimal control problems where the source/control is located outside the observation domain where the PDE is satisfied. The classical diffusion models lack this flexibility as they assume that the source/control is located either inside or on the boundary. This is essentially due to the locality property of the underlying operators. We use the nonlocality of the fractional operator to create a framework that now allows placing a source/control outside the observation domain. We consider the Dirichlet, Robin and Neumann source identification or optimal control problems. These problems require dealing with the nonlocal normal derivative (that we shall call interaction operator). We create a functional analytic framework and show well-posedness and derive the first order optimality conditions for these problems. We introduce a new approach to approximate, with convergence rate, the Dirichlet problem with nonzero exterior condition. The numerical examples confirm our theoretical findings and illustrate the practicality of our approach.where the source is either f (force or load) or z (boundary control) see [6,29,36]. In (1.1) there is no provision to place the source in Ω (cf. Figure 1). The issue is that the operator ∆ has "lesser 2010 Mathematics Subject Classification. 49J20, 49K20, 35S15, 65R20, 65N30.

show abstract

Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains

Cited by 21 publications

References 22 publications

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Adapted Numerical Methods for the Poisson Equation with $L^2$ Boundary Data in NonConvex Domains

External optimal control of nonlocal PDEs

Contact Info

Product

Resources

About