Cut finite element error estimates for a class of nonlinear elliptic PDEs

Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This is usually achieved via a geometric parameterization of its boundary via level-set functions. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form - Δ u +f_1(u)=f_2 with polynomial nonlinearity via the cut finite element method. Boundary conditions are enforced, using a Nitsche-type approach. To ensure stability and error estimates that are independent of the position of the boundary with respect to the mesh, the formulations are augmented with additional boundary zone ghost penalty terms. These terms act on the jumps of the normal gradients at faces associated with cut elements. A-priori error estimates are derived, while numerical examples illustrate the implementation of the method and validate the theoretical findings.

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1. Introduction

Fictitious domain methods have a long history, dating back to the pioneering work of Peskin [22] and are currently enjoying great popularity, having been successfully applied to a variety of problems. Several variants include such methods as the ghost-cell finite difference method [24], cut–cell volume method [21], immersed interface [17], ghost fluid [6], shifted boundary methods [2, 13, 20], –FEM [9], and CutFEM [3, 4, 5, 18, 11], among others. For a comprehensive overview of this research area, the interested reader is referred to the review paper [19]. Considerable impetus has been provided in the contexts of fluid–structure interaction and reduced order modeling for parametrically–dependent domains [12, 14].

Such cases pose severe challenges in the discretization and even result to simulations of diminished quality. For instance, the generation of a suitable conforming mesh is a challenging and computationally intensive task. As a means to bypass such complications, it is instructive to consider the actual computational domain of interest as being embedded in an unfitted background mesh. More precisely, this can be achieved via a geometric parameterization of its boundary via level–set geometries, using a fixed Cartesian background and its associated mesh for each new domain configuration. This approach avoids the need to remesh, as well as the need to develop a reference domain formulation. In such cases, immersed and embedded methods compare favorably to fitted mesh FEMs, providing simple and efficient schemes for the numerical approximation of PDEs in both cases of static and evolving geometries.

The overall objective of this note is to extend the a–priori analysis of cutFEM beyond the realm of linear problems. To this end, we propose an unfitted framework for the numerical solution of a semilinear elliptic boundary value problem with a polynomial nonlinearity. We start by introducing the model problem and the necessary notation in Section 2. Then, Section 3 focuses on the derivation of the a–priori error estimates and a numerical experiment is reported in Section 4, verifying the theoretical convergence rates and showcasing the accuracy of the method. The paper concludes with a brief discussion of our contributions and suggestions for future work in Section 5.

2. The model problem and preliminaries

As a model problem, we consider a semilinear elliptic boundary value problem of the form

(2.1)

where is a simply connected open domain with boundary . The nonlinearity is assumed to be of polynomial type . Such equations have been studied previously in the context of problems with critical exponents [8] and are referred to in the theory of boundary layers of viscous fluids [23] as Emden–Fowler equations. It is straightforward to verify that the weak formulation

(2.2)

of (2.1) admits a weak solution . Following a standard energy argument and assuming the force , the a–priori error bound

readily follows, indicating a continuous dependence of the solution on the data.

Implementation of an unfitted FEM for the discretization of (2.2) requires a fixed background domain which contains ; let its corresponding shape–regular mesh. The active mesh

is the minimal submesh of which covers and is in general unfitted to its boundary . As usual, the subscript indicates the global mesh size. The finite element space for discrete solutions will in fact be built upon the extended domain which corresponds to . Fictitious domain methods require boundary conditions at to be weakly satisfied through a variant of Nitsche’s method. On the other hand, coercivity over the whole computational domain is ensured by means of additional ghost penalty terms which act on the gradient jumps in the boundary zone; see, for instance, [5]. Therefore, a more detailed analysis of the interface grid is required; the submesh consisting of all cut elements is denoted

and the relevant set of faces upon which ghost penalty will be applied is given by

Considering the finite element space

for approximate solutions, we define discrete counterparts to the continuous bilinear and linear forms in (2.2), setting

(2.3)
(2.4)

for . Here,

denotes the outward pointing unit normal vector on the boundary

. The cutFEM discretization scheme reads as follows: find a discrete state , such that

(2.5)

for all , where the stabilization term

(2.6)

acts on the gradient jumps of over element faces in the interface zone and is included in the bilinear form to extend its coercivity from the physical domain to . The quantities and in (2.3) and (2.6) are positive penalty parameters; see Lemma 3.2 below.

3. Norms, approximation properties and a–priori analysis

The convergence analysis of the method (2.5) is based on the following mesh–dependent norms:

The trace inequality for and implies in particular:

A necessary approximation result is stated next:

Lemma 3.1 ([5], Lemma 5).

Let a linear –extension operator on , such that , , and

the Clément-type extended interpolation operator defined by

, where is the standard Clément interpolant. Then, the estimate

(3.1)

holds for every .

Regarding stability, the coercivity and continuity properties of the augmented bilinear form now read as follows:

Lemma 3.2 ([5], Lemmata 6 and 7).

Defining the method (2.5) with sufficiently large parameter and , then

(3.2)

for every , and

(3.3)

independently of and of the way in which the boundary intersects the background mesh.

Hence, due to the gradient penalty in the boundary zone, control of the –norm of the gradient may be extended over the whole active mesh .

We next quantify how the additional term affects the Galerkin orthogonality and consistency of the variational formulation (2.5).

Lemma 3.3 (Galerkin orthogonality).

Let be the solution to the semilinear problem (2.2) and its finite element approximation in (2.5). Then,

(3.4)
Proof.

Recalling the definitions of and in (2.3) – (2.4), it is immediate that the solution satisfies the equation , for every and the result follows. ∎

The following preliminary result investigating optimality with respect to interpolation is a key ingredient of our approach.

Proposition 3.4.

Let be the solution to the semilinear problem (2.2) and its finite element approximation in (2.5). Then, there exists a constant , independent of , , such that

(3.5)

where is the conjugate index of the power in the nonlinear term .

Proof.

Adapting for our purposes the procedure in the proof of [7, Thm. 5.3.3, p. 319] for the –Laplacian, a first observation is that there exists , such that

(3.6)

Then, denoting , we successively apply the coercivity estimate (3.2), (3.6) and the Galerkin orthogonality (3.4) to estimate

A bound for the leading two terms is readily implied by the continuity estimate (3.3), the Cauchy–Schwarz inequality and (3.1):

while the third term is estimated by

Hence, the assertion (3.5) already follows for .

Under some additional regularity requirements for the solution , we are now in a position to derive error estimates for our finite element approximations:

Theorem 3.5 (Optimal convergence).

Let be the solution to the semilinear problem (2.2) and its finite element approximation in (2.5). Then, .

Proof.

We first decompose the total error into its discrete–error and projection–error components; i.e.,

The desired estimate for the first term is already provided by (3.1). Hence, it suffices to prove the assertion for the latter, which is in turn bounded by Proposition 3.4. Indeed, by (3.1) and the properties of the Clément interpolant [10, p.69], estimate (3.5) yields

for . Recalling is the conjugate index of , clearly and the bound is optimal. ∎

4. Numerical validation

In order to verify the validity of the a-priori error estimate in Theorem 3.5

, numerical simulations have been implemented in a python environment, using the open–source Netgen/NGSolve-ngsxfem finite element software. We consider a two–dimensional test case of (

2.1) for with manufactured exact solution and right–hand side force defined by

in ; i.e., the unit disc centered at the origin. As in Section 2, the original domain is immersed in the background domain . To investigate orders of convergence, we consider a sequence of successively refined tessellations for with mesh parameters (). The stabilization constants , in (2.3) and (2.6) are taken to be equal to and respectively. By the theoretical error estimate stated in Theorem 3.5, we should expect first–order convergence rates with respect to the –norm and additionally second order for the –norm.

EOC EOC
7.74620e-2 2.47468e-3
3.90601e-2 0.988 5.83351e-4 2.085
1.93383e-2 1.014 1.33451e-4 2.128
9.63082e-3 1.006 3.34143e-5 1.999
4.80627e-3 1.003 8.12293e-6 2.040
2.40450e-3 0.999 2.01406e-6 2.012
Mean 1.002 2.049
Table 1. Errors and experimental orders of convergence (EOC) for and norms.

As illustrated in Table 1, the numerical findings validate the theoretically predicted rates of convergence and verify the effectiveness of the proposed framework.

5. Conclusions

The present note concentrated on the derivation of an a–priori error estimate for a cut finite element approximation of a semilinear model problem. To the authors’ best knowledge, this is one of the few instances in the literature that such an analysis has been carried out beyond a linear context. Our approach is based on classical arguments for the –Laplacian [7] and on key results from [5] for a stabilized unfitted method for the Poisson problem. Future work will delve more deeply in the analysis of unfitted FEMs for general time–dependent problems with nonlinearities. From a computational point of view, the effect of preconditioning on the performance of the method will be assessed in the spirit of [1, 16]. Finally, the method seems promising for controlling nonlinear PDEs with uncertainties, involving large deformations and/or topological changes [12, 14, 15].

Acknowledgments

This project has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No[1115] (PI: E. Karatzas), and the support of the National Infrastructures for Research and Technology S.A. (GRNET S.A.) in the National HPC facility - ARIS - under project ID pa190902.

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