Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Burman, Erik; Larson, Mats G.; Oksanen, Lauri

doi:10.1137/17m1163335

Cited by 15 publications

(31 citation statements)

References 34 publications

(52 reference statements)

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“…Proof. The first inequality in (11) has been proven in Lemma 5.1 in [24]. For the second inequality, see Lemma 5.2 of [23].…”

Section: The Stabilized Formulationmentioning

confidence: 90%

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A cut finite element method for the Darcy problem

Puppi¹

2021

Preprint

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We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is based on the Nitsche formulation studied in [12] and can be considered as a first attempt at extension in the unfitted case. The key feature is to add two ghost penalty operators to stabilize both the velocity and pressure fields. We rigorously prove our stabilized formulation to be well-posed and derive a priori error estimates for the velocity and pressure fields. We show that an upper bound for the condition number of the stiffness matrix holds as well. Numerical examples corroborating the theory are included.

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“…Proof. The first inequality in (11) has been proven in Lemma 5.1 in [24]. For the second inequality, see Lemma 5.2 of [23].…”

Section: The Stabilized Formulationmentioning

confidence: 90%

“…We mimic the proof of Proposition 2.1 in [11]. Let K ∈ T h (Ω I,h ) and K be the the reference element.…”

Section: Stability Estimatesmentioning

confidence: 99%

A cut finite element method for the Darcy problem

Puppi¹

2021

Preprint

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“…Herein we consider well-posed, but possibly indefinite advection-diffusion equations. However, the results extend to ill-posed advectiondiffusion equations using the ideas of [14] and [15].…”

mentioning

confidence: 80%

“…Remark 3.1. The constrained-minimization problem introduces an auxiliary variable, i.e., the Lagrange multiplier, which for stability reasons must be chosen as the discontinuous counterpart of the discretization space for the primal variable (unless stabilization is applied, see [14]). This results in a system with a substantially larger number of degrees of freedom than the standard Galerkin and the classical mixed method.…”

Section: The Model Problemmentioning

confidence: 99%

Primal Dual Mixed Finite Element Methods for Indefinite Advection-Diffusion Equations

Burman¹,

He²

2019

SIAM J. Numer. Anal.

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We consider primal-dual mixed finite element methods for the advection-diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy-and the L 2 -norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the H(div) norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.

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“…Several regularization techniques has been proposed to tackle Problem (1.1). Without being exhaustive, we may mention methods based on surface integral equations [12,23], Lavrentiev regularization [10,11], stabilized finite elements methods [15][16][17], quasi-reversibility method [13,18,26,35,36], fading regularization method [24,27], etc.…”

Section: Introductionmentioning

confidence: 99%

A dual approach to Kohn–Vogelius regularization applied to data completion problem

Caubet¹,

Dardé²

2020

Inverse Problems

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This paper focuses on a dual approach in order to study the data completion problem. A classical method to solve this problem is to minimize the so-called regularized Kohn-Vogelius functional. However this method needs to choose an appropriate parameter of regularization to ensure its efficiency in the numerical reconstruction. To avoid this difficulty, we propose to study the inverse problem through a dual problem. Using some well-chosen functional spaces and establishing theoretical results in a abstract setting, we prove the well-posedness of the dual minimization problem and the convergence of our regularized solution to the exact solution when the amount of noise on the data goes to 0. Moreover we prove that the regularized solution satisfies the well-known Morozov discrepancy principle. Then we establish that the minimization of the dual functional permits not only to stably obtain a good reconstruction of the missing data of the Cauchy problem but also to determine the value of a suitable parameter of regularization in the Kohn-Vogelius strategy. We finally present numerical results, in two and three dimensions, to underline the efficiency of the proposed method.

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Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Cited by 15 publications

References 34 publications

A cut finite element method for the Darcy problem

A cut finite element method for the Darcy problem

Primal Dual Mixed Finite Element Methods for Indefinite Advection-Diffusion Equations

A dual approach to Kohn–Vogelius regularization applied to data completion problem

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