An optimal order finite element method for elliptic interface problems

(1)

An optimal order finite element method for elliptic interface problems

Gunther H. Peichl1,∗and Rachid Touzani2,∗∗

1

University of Graz, Institute for Mathematics, Heinrichstr. 36, 8010 Graz, Austria. 2

Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand) and CNRS (UMR 6620), Campus Universitaire des Cézeaux, 63177 Aubière cedex, France.

A specific finite element method for problems involving interfaces is presented. The method allows for non fitted meshes and is well adapted for elliptic problems with jumps of coefficients along a closed curve. Error bounds for the presented method show optimal convergence.

1 Introduction

The numerical solution of boundary value problems involving interfaces requires some care in their numerical treatment. This is in particular the case when one considers free boundary problems formulated in a fixed domain like in level set formulations for instance. The main difficulty in such formulations is that the numerical solution experiences poor accuracy in the vicinity of the interface. To remedy such difficulties we make use of a fitted finite element method that we relax by using hybrid finite element techniques on the edges intersected by the interface. Let us mention related work in this area either using finite elements (e.g., [1, 3, 4, 6]) or finite differences (e.g., [2, 5]).

Let us define a model problem. Consider a bounded and regular domainΩ of R2with boundaryΓ and let γ denote a regular closed curve that separatesΩ into two bounded domains Ω+andΩ−, i.e.Ω = Ω+∪ γ ∪ Ω−. We define the boundary value problem:

− ∇ · (a∇u) = f inΩ,

u = 0 onΓ, (1)

where_{f ∈ L}2(Ω) and a is a function in L∞(Ω) such that a_|Ω± ∈ W1,∞(Ω±). In other words, we allow for possible jump of

a across γ. It is then well known that u|Ω± ∈ H2(Ω±), but clearly u /∈ H2(Ω).

2 A fitted finite element method

Let us consider a standard finite element approximation of Problem (1). Assuming thatΩ is polygonal, we denote by T_ha triangulation ofΩ and define the simplest finite element space

Vh= {v ∈ C0(Ω); v|K∈ P1∀ K ∈ Th, v|Γ = 0},

whereP₁is the space of affine functions inR2. A finite element approximation of Problem (1) is then defined by the variational problem:

Finduh∈ Vhsuch that

Ωa∇uh· ∇v dx =

Ωf v dx ∀ v ∈ Vh.

(2) The finite element approximation (2) is well known to result in poor accuracy, especially in the vicinity of the interface_γ. To avoid this difficulty, we first resort to a fitted finite element method. For this end, letE_hdenote the set of triangle edges and T_hγ the set of triangles that intersectγ. Let us next consider a piecewise affine approximation γhofγ, i.e. γhis a continuous closed curve inΩ which, restricted to a triangle, is a straight line and which coincides with γ if γ intersects a triangle along an entire edge.

Let us now assume for simplicity that the intersection ofγ with any triangle does not meet any vertex. For a triangle T ∈ Th, we splitT into one triangle K₁T sharing one edge withγhand one quadrilateral that we split again into two subtrianglesK₂T andK₃T. Let thenT_Tγ = {K₁T, K₂T, K₃T} and let E_hγ stand for the set of edges of the triangulation that intersect the interface γ. The fitted finite element mesh is defined as

TF h = Th∪ T ∈Tγ h ∪K∈Tγ T K . ∗ _{E-mail: gunther.peichl@kfunigraz.ac.at,}

∗∗ _{Corresponding author} _{E-mail: Rachid.Touzani@univ-bpclermont.fr, Phone: +33 473 407 706, Fax: +33 473 407 064}

PAMM · Proc. Appl. Math. Mech. 7, 1025403–1025404 (2007) / DOI 10.1002/pamm.200701144

(2)

The next step consists in defining an approximationah ofa that admits a jump across γh rather thanγ. This issue is described in [7]. Let us now define the space

Wh= Vh+ Xh where Xh= {v ∈ C0(Ω); v|Ω\Sγ

h ∈ P1∀ K ∈ T

γ

T, ∀ T ∈ Thγ}. HereS_hγ is the union of triangles intersected byγ. The fitted finite element approximation is defined by:

FinduF_h ∈ Whsuch that Ωah∇u F h · ∇v dx = Ωf v dx ∀ v ∈ Wh. (3) Assuming the regularity property_{u ∈ W}2,∞(Ω+∪ Ω−) we can prove the error bound

∇(u − uF

h)L2_(Ω)2≤ Ch u_W2,∞_(Ω+_∪Ω−₎.

3 A hybrid finite element approximation

Problem (3) seems to give the right answer to the accuracy issue in the numerical solution of elliptic interface problems by finite element methods. It appears however that, in view of an iterative process like in fixed mesh free boundary problems, its practical efficiency is not guaranteed. To overcome this difficulty, it is possible to relax the continuity condition on the added degrees of freedom by using a hybrid formulation. We replace then the spaceXhby

Yh= {v ∈ L2(Ω); v|Sγ

h, v|K∈ P1∀ K ∈ T

γ

h, [v] = 0 on e, ∀ e ∈ Eh\ Ehγ}. We also define_Z_h= V_h+ Y_hand the space in which the corresponding Lagrange multipliers lie:

Qh= {μ ∈

e∈Eγ

h

; μ|e= const. ∀ e ∈ Ehγ}. The new formulation is thus defined by the variational problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Find(uH_h_{, λ}_h) ∈ Z_h× Q_hsuch that Ωah∇u H h · ∇v dx − e∈Eγ h eλh[v] ds = Ωf v dx ∀ v ∈ Zh, e∈Eγ h eμ[u H h] ds = 0 ∀ μ ∈ Qh.

Based on an inf-sup condition, we prove in [7] the following error bound

∇(u − uH

h)L2_(Ω)2 ≤ Ch u_W2,∞_(Ω+_∪Ω−₎

if the regularity propertyu ∈ W2,∞(Ω+∪ Ω−) holds.

Let us finally mention that numerical experiments reported in [7] confirm these theoretical bounds and give the expected error behaviour inL2andL∞norms.

References

[1] T. Belytschko, N. Mo¨es, S. Usui, C. Parimi, Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50(4), 993-1013 (2001).

[2] F. Bouchon, G. Peichl, A Second order immersed interface technique for an elliptic Neumann problem, Num. Methods for PDE, 23, 400-420 (2007).

[3] P. Hansbo, C. Lovadina, I. Perugia, and G. Sangalli, A Lagrange multiplier method for the finite element solution of elliptic interface problems using non?matching meshes, Numer. Math., 100, 91–115 (2005).

[4] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191, 5537–5552 (2002).

[5] Z. Li, T. Lin, X. Wu, New cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96, 61–98 (2003).

[6] B.P. Lamichhane, B.I. Wohlmuth, Mortar finite elements for interface problems, Computing, 72, 333–348 (2004). [7] G. Peichl, R. Touzani, An accurate finite element method for elliptic interface problems, Submitted.