A posteriori error estimates for elliptic problems with Dirac delta source terms

Araya, Rodolfo; Behrens, Edwin M.; Rodrı́guez, Rodolfo

doi:10.1007/s00211-006-0041-2

Cited by 51 publications

(81 citation statements)

References 12 publications

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“…Consequently, a refined analysis based on L p spaces is required. This can again be done along the lines of [3]; cf. also [19].…”

Section: Application To the Sipg Methodsmentioning

confidence: 99%

“…If t = 2, we simply write H s (Ω) = W s,2 (Ω). Following [3], Section 2, the above weak formulation is well-posed.…”

Section: )mentioning

confidence: 99%

“…Consequently, the numerical approximation of (1.1)-(1.2) by, for example, finite element methods, requires a non-standard analysis. Here, in the context of conforming FEM, we mention the a priori results in [7,17], as well as the a posteriori error analysis in [3]. For DG approximations to low-regularity problems, see, e.g., [13,19].…”

Section: )mentioning

confidence: 99%

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Discontinuous Galerkin methods for problems with Dirac delta source

Houston

Wihler

2012

ESAIM: M2AN

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Abstract. In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L 2 -norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.Mathematics Subject Classification. 65N30.

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“…Consequently, a refined analysis based on L p spaces is required. This can again be done along the lines of [3]; cf. also [19].…”

Section: Application To the Sipg Methodsmentioning

confidence: 99%

“…If t = 2, we simply write H s (Ω) = W s,2 (Ω). Following [3], Section 2, the above weak formulation is well-posed.…”

Section: )mentioning

confidence: 99%

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Discontinuous Galerkin methods for problems with Dirac delta source

Houston

Wihler

2012

ESAIM: M2AN

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“…In this section we define a residual error estimator by combining ideas from [17] and [2] for advection-reaction-diffusion problems with those in [3] for problems with a delta source term.…”

Section: Remarkmentioning

confidence: 99%

“…In all these works smooth source terms are considered. On the other hand, an a posteriori error analysis has been recently developed in [3] for the Laplace equation with a delta source term. To the best of the authors knowledge, no a posteriori error analysis has been performed for the advection-diffusion-reaction equation with a non regular right hand side.…”

Section: Introductionmentioning

confidence: 99%

An adaptive stabilized finite element scheme for a water quality model

Araya

Behrens

Rodrı́guez

2007

Computer Methods in Applied Mechanics and Engineering

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Residual type a posteriori error estimators are introduced in this paper for an advection-diffusion-reaction problem with a Dirac delta source term. The error is measured in an adequately weighted W 1,p -norm. These estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.

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Numerical analysis for the approximation of optimal control problems with pointwise observations

Chang

Gong

Yan

2013

Math Methods in App Sciences

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Communicated by Q. WangIn this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results.of elliptic equations with measure data presented in [16] and [17], a priori error estimates for the control, the state, and the adjoint state are derived. Moreover, the a posteriori error estimates of residual type are presented, which can be applied as indicators for the mesh refinement in adaptive finite element methods. Numerical examples are also included to illustrate the theoretical results. It should be pointed out that our theoretical and numerical results are still valid when the penalty parameter˛is very small, which often appears in many applications, for example, inverse problems.Our paper is organized as follows. In the next section, we present the mathematical setting and formulate the optimal control problems in detail. Finite element approximation to optimal control problems is also presented. In Sections 3 and 4, we prove the a priori error and a posteriori error estimates, respectively. We finally present numerical results, which confirm our analytical findings in section 5.where B is the linear operator from U to L 2 . /, U is the control space. Here, B can be an identity operator such that Bu D u or Bu D ! u where ! is an open subset of and ! the characteristic function of the subset !. z D .z 1 , , z m / 2 R m is given, and X j (j D 1, , m) are space points strictly contained in . Moreover, U ad U is a closed convex subset of typeAccording to the assumptions on and A, for each Bu 2 L 2 . /, we have y.x; u/ 2 H 2 . / \ H 1 0 . /, thus y.x; u/ 2 C. / by imbedding theorem and the aforementioned optimal control problem is well-defined. From standard arguments ([18]), we can prove that the

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A posteriori error estimates for elliptic problems with Dirac delta source terms

Cited by 51 publications

References 12 publications

Discontinuous Galerkin methods for problems with Dirac delta source

Discontinuous Galerkin methods for problems with Dirac delta source

An adaptive stabilized finite element scheme for a water quality model

Numerical analysis for the approximation of optimal control problems with pointwise observations

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