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Inverted finite elements approximation of the Neumann problem for second order elliptic equations in exterior two-dimensional domains

R Belbaki, S K Bhowmik, T Z Boulmezaoud, N Kerdid, S Mziou

TL;DR

This work extends the Inverted Finite Elements Method to second-order elliptic equations in two-dimensional exterior domains under Neumann boundary conditions, accommodating coefficients that vary to infinity. A robust weighted variational framework is developed, ensuring well-posedness via a mean-zero constraint and Hardy-type coercivity. The IFEM discretization uses a domain decomposition, a polygonal inversion, and a graded mesh to transform the unbounded region to a bounded star domain, enabling accurate and efficient computation; a priori error estimates verify convergence, and numerical tests in MATLAB confirm robustness to coefficient growth and mesh gradation. The approach yields reliable solutions in exterior Neumann problems without requiring explicit far-field representations, with practical impact for problems in unbounded domains and variable coefficients.

Abstract

We use inverted finite elements method for approximating solutions of second order elliptic equations with non-constant coefficients varying to infinity in the exterior of a 2D bounded obstacle, when a Neumann boundary condition is considered. After proposing an appropriate functional framework for the deployment of the method, we analyse its convergence and detail its implementation. Numerical tests performed after implementation confirm convergence and high efficiency of the method.

Inverted finite elements approximation of the Neumann problem for second order elliptic equations in exterior two-dimensional domains

TL;DR

This work extends the Inverted Finite Elements Method to second-order elliptic equations in two-dimensional exterior domains under Neumann boundary conditions, accommodating coefficients that vary to infinity. A robust weighted variational framework is developed, ensuring well-posedness via a mean-zero constraint and Hardy-type coercivity. The IFEM discretization uses a domain decomposition, a polygonal inversion, and a graded mesh to transform the unbounded region to a bounded star domain, enabling accurate and efficient computation; a priori error estimates verify convergence, and numerical tests in MATLAB confirm robustness to coefficient growth and mesh gradation. The approach yields reliable solutions in exterior Neumann problems without requiring explicit far-field representations, with practical impact for problems in unbounded domains and variable coefficients.

Abstract

We use inverted finite elements method for approximating solutions of second order elliptic equations with non-constant coefficients varying to infinity in the exterior of a 2D bounded obstacle, when a Neumann boundary condition is considered. After proposing an appropriate functional framework for the deployment of the method, we analyse its convergence and detail its implementation. Numerical tests performed after implementation confirm convergence and high efficiency of the method.
Paper Structure (5 sections, 3 theorems, 63 equations, 4 figures, 3 tables)

This paper contains 5 sections, 3 theorems, 63 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Function spaces and variational formulation.
  3. Inverted Finite Element Method. Discretization.
  4. Implementation and computational results
  5. Conclusion

Key Result

Proposition 2.1

Under assumptions (${\mathscr H}_{1}$)-(${\mathscr H}_{2}$), system second_ord_equa-neumann_code-mean_cond has one and only one solution $u \in W^{1}_{{{\rm log}}}({\omega}_{\rm ext})$. Moreover, for some constant $C > 0$ depending only on $\omega$ and the coefficient $\sigma$.

Figures (4)

  • Figure 1: (left) decomposition of the exterior domain ${\omega}_{\rm ext}$ into the union of four infinite triangles and a bounded domain ${\Omega}_0$. (right) The four triangles $S_1, \cdots, S_4$ make up the fictitious bounded domain $\Omega_\star$.
  • Figure 2: (example 1) global relative weighted $L^2$ error on $u$ (left figure) and global relative $L^2$ error on $\nabla {u}$ (right figure).
  • Figure 3: (example 1) exact solution (left) and approximate solutions (right) with $\mu = 1$.
  • Figure 4: (exemple 2) global relative weighted $L^2$ error on $u$ (left figure) and global relative $L^2$ error on $\nabla {u}$ (right figure) versus $h$.

Theorems & Definitions (7)

  • Proposition 2.1
  • proof : Proof of Proposition \ref{['proof_propo_exist']}
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemma_prth']}
  • Remark 3.1
  • Proposition 3.1
  • proof