Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems

Dolejší, Vít; Roskovec, Filip

doi:10.21136/am.2017.0173-17

Cited by 12 publications

(22 citation statements)

References 25 publications

(40 reference statements)

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“…where r h (z h )(·) denotes the residual of the primal problem given by (10). Similarly, exploiting in addition the Galerkin orthogonality (14) and relation (15), we get the dual error identity…”

Section: Abstract Goal-oriented Error Estimatesmentioning

confidence: 99%

On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

Dolejší

Tichý

2020

J Sci Comput

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We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The goal-oriented error estimates require the solution of the primal as well as dual algebraic systems. We solve both systems simultaneously using the bi-conjugate gradient method which allows to control the algebraic errors of both systems. We develop a stopping criterion which is cheap to evaluate and guarantees that the estimation of the algebraic error is smaller than the estimation of the discretization error. Using this criterion and an adaptive mesh refinement technique, we obtain an efficient and robust method for the numerical solution of PDEs, which is demonstrated by several numerical experiments.Keywords Goal-oriented error estimates · algebraic errors · BiCG method · adaptivity Mathematics Subject Classification (2010) 65N15 · 65N30 · 65F10 · 15A06 IntroductionLet Ω ⊂ R d be a polygonal domain with the boundary Γ = ∂Ω. We consider an abstract partial differential equation in the form

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Section: Abstract Goal-oriented Error Estimatesmentioning

confidence: 99%

On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

Dolejší

Tichý

2020

J Sci Comput

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“…Whilst this approach is shown to be effective in Subsection 4.3, it implies an additional computational cost associated with solving the adjoint equation on a mesh with four times as many elements. In future work, a more efficient approach will be implemented, such as the one described on pp.590-593 of [10], which solves local PDEs to approximate q * .…”

Section: Goal-oriented Error Estimatementioning

confidence: 99%

“…Modifying (10) to account for a set of tidal turbines T amounts to choosing an appropriate drag coefficient C d . Suppose turbine T has thrust coefficient c T , area A T and footprint indicated by 1 T .…”

Section: Tidal Turbine Modellingmentioning

confidence: 99%

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Goal-Oriented Error Estimation and Mesh Adaptation for Shallow Water Modelling

Wallwork¹,

Barral²,

Kramer³

et al. 2019

Preprint

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Numerical modelling frequently involves a diagnostic quantity of interest (QoI) - often of greater importance than the PDE solution - which we seek to accurately approximate. In the case of coastal ocean modelling the power output of a tidal turbine farm is one such example. Goal-oriented error estimation and mesh adaptation can be used to provide meshes which are well-suited to achieving this goal, using fewer computational resources than required by other methods, such as uniform refinement.A mixed discontinuous/continuous Galerkin approach is applied to solve the nonlinear shallow water equations within the Thetis coastal ocean finite element model. An implementation of goal-oriented mesh adaptation is outlined, including an error estimate which takes account of the discontinuities in the discrete solution and a method for approximating the adjoint error. Results are presented for simulations of two model tidal farm configurations. Convergence analysis indicates that the anisotropic goal-oriented adaptation strategy yields meshes which permit accurate QoI estimation using fewer computational resources than uniform refinement.

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“…Their idea is to split the error into two components where the first error is bounded by a computable dual-weighted residual and the second one, claimed small in [5], is estimated via equilibrated fluxes. An important focus whose rigorous investigation has been started only recently is the theory for nonconforming, discontinuous Galerkin, and mixed methods: let us cite in particular Mozolevski and Prudhomme [34] and Dolejší et al [13,15]. Finally, to the best of our knowledge, with the exception of [13,32,[42][43][44], all the above-cited results rely on the assumption that both the primal and the dual discrete problems are solved exactly.…”

Section: Introductionmentioning

confidence: 99%

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

Mallik

Vohralı́k

Yousef

2020

Journal of Computational and Applied Mathematics

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We derive a unified framework for goal-oriented a posteriori estimation covering in particular higher-order conforming, nonconforming, and discontinuous Galerkin finite element methods, as well as the finite volume method. The considered problem is a model linear second-order elliptic equation with inhomogeneous Dirichlet and Neumann boundary conditions and the quantity of interest is given by an arbitrary functional composed of a volumetric weighted mean value (source) term and a surface weighted mean (Dirichlet boundary) flux term. We specifically do not request the primal and dual discrete problems to be resolved exactly, allowing for inexact solves. Our estimates are based on H(div)-conforming flux reconstructions and H 1-conforming potential reconstructions and provide a guaranteed upper bound on the goal error. The overall estimator is split into components corresponding to the primal and dual discretization and algebraic errors, which are then used to prescribe efficient stopping criteria for the employed iterative algebraic solvers. Numerical experiments are performed for the finite volume method applied to the Darcy porous media flow problem in two and three space dimensions. They show excellent effectivity indices even in presence of primal and dual algebraic errors and enable to spare a large percentage of unnecessary algebraic iterations.

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Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems

Cited by 12 publications

References 25 publications

On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

Goal-Oriented Error Estimation and Mesh Adaptation for Shallow Water Modelling

Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers

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