Analysis of the fictitious domain method with penalty for elliptic problems

The fictitious domain method with H1-penalty for elliptic problems is considered. We propose a new way to derive the sharp error estimates between the solutions of original elliptic problems and their H1-penalty problems, which can be applied to parabolic problem with moving-boundary maintaing the sharpness of the error estimate. We also prove some regularity theorems for H1-penalty problems. The P1 finite element approximation to H1-penalty problems is investigated. We study error estimates between the solutions of H1-penalty problems and discrete problems in H1 norm, as well as in L2 norm, which is not currently found in the literature. Thanks to regularity theorems, we can simplify the analysis of error estimates. Due to the integration on a curved domain, the discrete problem is not suitable for computation directly. Hence an approximation of the discrete problem is necessary. We provide an approximation scheme for the discrete problem and derive its error estimates. The validity of theoretical results is confirmed by numerical examples.

the fictitious domain is discretized by a uniform mesh, independent of the original boundary. The advantage of this approach is that we can avoid the time-consuming construction of a boundary-fitted mesh. One of these approaches is the penalty fictitious domain method which is based on a reformulation of the original problem in the fictitious domain by using a penalty parameter (see [2] for an introduction of other kinds of fictitious domain methods). In this article, we consider only the fictitious domain method with penalty. Obviously, the fictitious domain method is of use for time-dependent moving-boundary problems. Although there exist some ways to derive the sharp error estimates for elliptic problems (cf. [8,11,12]), it seems none of them has been applied to parabolic problem such that the sharpness of the error estimates are maintained. Our motivation lies in the study of the penalty fictitious domain method which can be applied to these time-dependent moving-boundary problems maintaining the sharpness of the error boundary. This is of obvious importance, and it seems that little is known in this direction. The fictitious domain method with penalty for parabolic problem firstly appeared in [6] to prove the existence of the solution for parabolic problem in time-dependent domain. Then, in [7], the convergence and finite difference approximation is given, but without error estimates. The H 1 -penalty parabolic problem equals to a special interface problem, and in [1] the error estimate for elliptic and parabolic interface problem is studied. However, it is not so suitable to the H 1 -penalty problem, and still, is only for time-independent domain when considering parabolic interface problem. As a primary step towards this final end, herein we examine some new methods of error analysis for elliptic problems that can be easily applied to parabolic problems in time-dependent domain with sharp error estimate (which has been presented in our another paper [14]). This is the purpose of this paper.
In order to illustrate our results, we consider the Dirichlet boundary value problem for the Poisson equation. The weak form (Q) reads as Find u ∈ H 1 0 ( ) such that (∇u, ∇v) = ( f, v) , ∀v ∈ H 1 0 ( ), (1.1) where ⊂ R 2 denotes a smooth bounded domain, (·, ·) is the inner product of L 2 ( ) and f ∈ L 2 ( ). We can find a rectangular domain D ⊃ , 1 = D\ , and turn to solve the H 1 -penalty problem (Q ) with penalty coefficient 0 < 1, wheref is the zero extension of f into D.
Another example of applying the fictitious domain method with penalty to (1.1) is, which we call the L 2 -penalty problem. This is of interest; however, the L 2 -penalty problem is beyond the scope of this paper, in which we shall concentrate our attention to the H 1 -penalty problem. We have presented some results of L 2 -penalty method in [15].
The error u − u 1, ( · 1, = · 0, + | · | 1, ) has been analyzed by many authors, where · 1, is the H 1 ( ) norm. In [7], it is bounded by C √ (C is some constant, so as in the following), and in [8,11,12] the sharp estimate C is achieved. In this paper, we give a new way to derive the sharp estimate. Moreover, we present some regularity analysis of u , which is useful for studying the H 1 and L 2 error between the solutions of H 1 problem (Q ) and its discrete problem, which is denoted as (Q ,h ).
A Cartesian mesh can be introduced to the rectangular domain D to get a uniformed triangulation T h , h is the maximum diameter of the triangles of T h . V h (D) is the subspace of all piecewise linear continuous functions subordinate to T h . Then (Q ,h ) reads as: Although there exist many works on finite element error estimate for elliptic problem with discontinuous coefficient or boundary unfitted mesh, we notice the discontinuous coefficient of H 1 -penalty problem is dependent on the parameter , such that methods on those works may not be so suitable for our problem. And most of those are not easy to apply to parabolic problem. So, we give a new analysis of the error estimate.
In the literature, there are several works devoted to the study of the H 1 error between u ,h and u in . For example, in [12], it is proved that the H 1 error is bounded by In our work, we prove a similar result with the analysis with a much simpler method of the analysis. The analysis of [12] is to consider the estimate of u ,h − u 0 − u 1 to derive the final estimate of u − u ,h , where u 0 is the zero extension of u, and u 1 is the solution of problem: Here, n is the unit outward normal to viewed as a boundary of , and n − is opposite to n. We found the analysis in [12] to be complicated and not directly. Our method for estimating of u − u ,h is simpler, which is to find some interpolation of u , denoted as v h , and then estimate u − v h by using a regularity theorem of (Q ). Moreover, we show the L 2 error is bounded by C( + h + √ h). A similar result of H 1 error for the elliptic problem in a specific domain is given in [8].
In discrete problem (Q ,h ), we notice that we have to calculate the inner-product in a curved domain, for example, (∇u ,h , ∇v h ) . So the discrete problem cannot be directly computed. We find that few prior works have provided a sufficient discussion on this issue; however, it is necessary to give an approximation scheme for solving (Q ,h ) and the associated error estimates when applying the finite element method to computation.
Herein, we present an approximation scheme, that is, instead of solving (Q ,h ) we solve some problem (Q ,h ) approximating to (Q ,h ). (Q ,h ) reads as: whereˆ is a polygon approximating to , and f h is some interpolation off . With In above, we have restricted our attention to Dirichlet boundary problem. For Neumann and mixed boundary problems we also consider the approximation of H 1 -penalty problems. Further, for Neumann boundary problem, the discrete problem is investigated. Although some results we obtain are similar to those of [12], as we mentioned in the beginning, our work is focus on solving parabolic problems with time-dependent domain, and all methods of analysis are applicable to this class of problems.
The rest of this paper is organized as follows. The H 1 -penalty problems for original problems with Dirichlet, Neumann, and mixed boundaries are given in Sect. 2, as well as the analysis of error estimates between the solutions of original problems and H 1penalty problems, in a different way from that in [8,11].
In Sect. 3, we present some regularity theorems for H 1 -penalty problems. The H 1penalty problem is in a sense equivalent to a kind of interface problem. The regularity theorem for the interface problem has been studied in [9]. However, we make several improvements in priori estimates and identify some higher-order regularity for our problems. The theorems will be used in Sect. 4 to make the error estimate more simple than that of [12].
The Sect. 4 is devoted to discrete problems. Finite element approximations are investigated. Using the same separation method of the triangulated domain as in [12], regularity theorems in Sect. 3 and some lemmas from [3,10], we obtain the error estimates in H 1 norm with the same order as in [8,12]. Moreover, we give the higherorder L 2 norm error estimates.
We consider a scheme approximating to the discrete problem in Sect. 5. We introduce a new discrete problem (Q ,h ) to approximate to the discrete problem (Q ,h ). Of necessity, due to insufficient prior reported works on this issue in the literature, we derive some error estimates of the scheme to make the numerical analysis of the fictitious domain method with H 1 -penalty more complete.
Finally, we give some numerical experiments to verify our theoretical results in Sect. 6.

H 1 -penalty problems of fictitious domain method for elliptic problems
Following the notation given in the previous section, we state the H 1 -penalty problem for the original elliptic problem with homogeneous Dirichlet, Neumann and mixed boundary conditions. In addition, we write = ∂ .

Dirichlet boundary value problem
First, we consider the Dirichlet boundary value problem (1.1) and its H 1 -penalty problem (1.2).

Theorem 2.1
There exist unique solutions u and u for (1.1) and (1.2), respectively, and we have the following estimates: Those error estimates themselves are not new: they have been stated in [8] and [11]. The main process to prove those estimates in [8] and [11] are different (see Remark 3 below). We shall give a somewhat new proof which will be used for parabolic problems. Before stating that, we recall the well-known extension and trace theorems that we frequently use.
The following lemma is a readily obtainable consequence of Theorem 8.3 of Chapter 1 in [5].
Proof of Theorem 2.1 Firstly, by the Lax-Milgram Theorem, the unique existence of u and u is obvious. And we can obtain the estimate for the solution of the H 1 -penalty problem (1.2), Without loss of generality, we assume 0 < which leads to an estimate of u | 1 , in particular, Therein, the first inequality is deduced by Friedrichs' inequality, and the second term of the right-hand side is bounded by C 2 f 0 u 1, 1 . Thus we have Next, we consider the trace operators We define an operator A : We observe that for any v ∈ H 1 0 ( ), which completes the proof.

Remark 2
The conclusion of Theorem 2.1 remains valid even for f ∈ H −1 ( ).

Remark 3
The saddle-point method in [8] requires a symmetric variational form, and the operator method in [11] is not so easy to deal with the operator of time-derivative when one considers parabolic problems. However, our method of analysis is applicable to parabolic problems; this is a recent achievement, and will be presented in our future work.

Neumann boundary value problem
The original problem (Q) with homogeneous Neumann boundary condition reads as: The H 1 -penalty problem (Q ) reads as: (2.4)

Theorem 2.4 There exist unique solutions u and u for (2.3) and (2.4)
, respectively, and we have the following estimates: Proof Firstly, substituting v = u into (2.4), we obtain This gives Setting v = u − E 1,0 ( )u in (2.7), and noticing that v| = 0, we have And we see that Together with Friedrichs' inequality, we have Thus, we proved (2.6).
In order to derive (2.5), we . Then, together with (2.6), we have which implies (2.5). We complete the proof.

Remark 4
Recall Remark 3 again. It should be noticed that our analysis method is much simpler than that in [8,11].

Mixed boundary value problem
The domian D\ is assumed to be split into two part 1 and 2 , which respectively share the boundaries 1 and 2 with . In addition both share non-empty measure boundary with D (see Fig. 1). Set The original problem (Q) with homogeneous mixed boundary is stated as: (2.8) The H 1 -penalty problem (Q ) reads as: (2.9) Theorem 2.5 There exist unique solutions u and u for (2.8) and (2.9), respectively, and we have the following estimates: (2.12) Proof The following results have already been achieved (Theorem I-8 in [7]): . From (2.13) and (2.14), we have (2.16) We find that v| * = 0 and substitute this v into (2.9) to obtain This gives Combining with u 1, 2 ≤ C|u | 1, 2 and (2.16), it shows (2.12).
from which we obtain that Next, we define a continuous operator A : It follows from the Lax-Milgram theorem that and the proof is completed.
Remark 5 Basically, the proof for the mixed boundary case is a combination of those for the Dirichlet and Neumann boundary cases.
Remark 6 All of the above in this section, which involve homogeneous boundary conditions are also suitable for the non-homogeneous boundary value problems.

The regularity of the solutions of H 1 -penalty problems
This section is devoted to the regularity theorems for (Q ) with homogeneous Dirichlet, Neumann, and mixed boundary conditions respectively. As it is shown in [7], these (Q ) are equal to certain interface elliptic problems, denoted as (P ). There are some regularity theorems for the interface elliptic problems in the literature (see [1,9] for example), however, these are not specific to our problems. Our particular objective is to deduce explicit dependence of various norms of u on the penalty parameter . We show some estimates which are only suitable for our problems, as well as some higher-order regularity which will be used in the study of the H 1 -penalty parabolic problem.

Dirichlet boundary value problem
As a first step, let us assume D is sufficiently smooth. Then, we have the following theorem.
First, we recall the following basic regularity result:

Lemma 3.2
Let ω ⊂ R 2 be a bounded domain. Assume that the boundary ∂ω is divided into two disjoint smooth components ∂ω 1 and ∂ω 2 ; .
Before the proof, we see that, by applying Green's formula, (1.2) is equivalent to (P ), which reads as: Then, applying Lemma 3.2, we obtain Now, we can state the following proof.
Proof of Theorem 3.1 From the discussion above, we only need to show that u | ∈ H 2 ( ) and u 2, ≤ C. This is a well-known result; however, we want to present a brief proof here, because we will show that, by a slight change of this process, we can obtain a higher-order regularity, with smoother assumptions on f .
Since (θ f − ∇u ∇θ − ∇(u ∇θ))| ∈ L 2 ( ), obviously, we have So, u 2 ∈ H 2 (U ) and u 2 2,U ≤ C. where Letũ 2 be the zero extension of u 2 onto R 2 . Substituting For i = 1, 2, we consider any sequence h j → 0, as j → ∞. We see that h jũ 2 converges weakly to some function φ ∈ L 2 (R 2 ), and also in the sense of distribution. Consequently, D i D 1ũ2 = φ ∈ L 2 (R 2 + ). This shows that all second derivatives, except D 2 2 u 2 , are in L 2 (R 2 ). We find that the equation of u 2 is also equivelent to As f 3 0,D ≤ C and g 1 2 , and from the equation of u 1 above, w satisfies which comes from Green's formula.

Theorem 3.3 Under the assumption that f ∈ H k ( ), for all non-negative integers k, we have
(3.5)

Remark 8
In the above two theorems, we both assume that D is sufficiently smooth; however, in our case, D is a rectangle (a convex polygon). From the discussion in [3,4] on elliptic problems in non-smooth domains, we can keep Theorem 3.1 remains true for any convex polygon D.

Neumann boundary value problem
As a first step, let us assume D is sufficiently smooth. Then, we have the following theorem.

Theorem 3.4 (Q) is the original problem with homogeneous Neumann boundary. f ∈ L 2 ( ), then the corresponding H 1 -penalty problem (Q ) has a unique solution
Before the proof, we show that, by applying the Green's formula, (Q ) is equivalent to (P ), which reads as: Since the right-hand-side function is 0, and with the homogeneous Dirichlet boundary of D, it concludes that u | 1 ∈ H 2 ( 1 ) and This means we have left to prove only u | ∈ H 2 ( ).
Proof The process of the proof is very similar to that of Theorem 3.1. In fact, we only need to replace the 1 in the proof of Theorem 3.1 by .

Remark 9
The same comments as those in the Dirichlet case apply here, specifically, we can obtain higher-order regularity for f ∈ H k ( ), and, if D is a convex polygon, Theorem 3.4 remains true.

Mixed boundary value problem
The (Q ) for original problem with homogeneous mixed boundary is equivalent to the problem (P ): where 3 is the common boundary of 1 and 2 . By an analogue of the previous proof, we can obtain that u | ∈ H 2 ( ).

Finite element approximation and discrete problems
Recall that the Cartesian mesh is introduced to the rectangular domain D to get a uniform triangulation T h , and h is the maximum diameter of the triangles of T h . Each K ∈ T h is assumed to be a closed set. V h (D) ⊂ H 1 0 (D) is the subspace of all piecewise linear continuous functions subordinate to T h .

Dirichlet boundary value problem
We consider the discrete problem (1.3). (1.3). u is the solution of (1.2), and we have

Lemma 4.1 There exists a unique solution u ,h ∈ V h (D) for
Then we find Applying the Poincaré inequality to the left-hand-side, we have Thus, we have proved the result.
To estimate we need some lemmas, which can be found in [12], and several other similar results in [10,13]. For a curve γ in C 2 (R 2 ) and δ > 0, we define a δ-neighborhood If we assume v ∈ H 2 (R), then we have

Lemma 4.3 Suppose w ∈ H 2 (D), and we define I K w as the linear interpolation of w on the vertices of a triangle K ∈ T h . Then, we have
where ν i , i = 1, 2, 3, are vertices of K . (A B means that there exist constants depending on the regularity of the triangulation C 1 , Before the proof, we define some notations: We may assume that T \T 0 = ∅ and T 1 \T 1 = ∅ without loss of generality.

Proof of Theorem 4.4 We define
for all others vertices ν, and substitute this v h into the right-hand-side of (4.1). We find that u − v h 1, 1 = u 1, 1 ≤ C . To estimate u − v h 1, , we use the scheme proposed in [12] by using Lemmas 4.2 and 4.3. However, there are several differences between our analysis and that of [12], because we apply our regularity theorem presented in the previous section, which simplifies the analysis.
For every K ∈ T 0 , we have We want to show that v h 0,K ≤ C I K u 1,K . There are two possibilities: (1) if for all ν i = ν K i ∈ (K ), i = 1, 2, 3, u (ν i ) have the same sign, then obviously, and v h 0,K ≤ C I K u 0,K ; (2) if we consider the case where for ν i , i = 1, 2, 3, u i = u (ν i ) do not all have the same signs, then, without loss of generality, we assume that |u 1 | = u ∞,K and u 1 u 2 ≥ 0 and u 3 ≤ 0. We have ∇( Thus, we have v h 0,K ≤ (1 + h K ) I K u 1,K , which gives v h 1,K ≤ |v h − I K u | 1,K + C I K u 1,K ≤ |v h − I K u | 1,K + C I K u − u 1,K + C u 1,K (4.6) By the standard interpolation error estimates, we have We notice that there exists Next, we set By definition, we see that (K ) = (K ) ∪ (K ), (K ) = ∅, (K ) = ∅. There are two possibilities: for (See Figs. 2 and 3.) At this stage, we apply the Sobolev and Morrey's inequalities. Let ω be a Lipschitz domain in R 2 . They are given as We choose q = 4, and define K = K ∩ , K 1 = K ∩ 1 , and we have Hence, we obtain |v h | 1,K ≤ C u 2,K 1 + Ch 1 2 u 2,K . Then, applying the same trick to (ii), we can show that Combining (i) and (ii), we have where the last inequality is from Theorem 3.1. Hence, we get Recalling that u − v h 1, 1 = u 1, 1 ≤ C and other estimates from the beginning of the proof, we have Hence, the theorem follows from Lemma 4.1.

Remark 10
Since we have u 2, 1 ≤ C , other choices for v h than that above can be taken, such as whereū is the extension of u | 1 onto D with ū 2,D ≤ C u 2, 1 , and the estimate result still holds.
To estimate u − u ,h 0,D , we need the adjoint boundary value problem, which reads as: For any given f ∈ L 2 ( ), find u f ∈ H 1 0 (D) such that (∇v, ∇u f ) The last inequality follows from Theorems 3. 1 and 4.4.
With u 1, 1 ≤ C and u ,h 1, 1 ≤ C , we have proved the result.

Neumann boundary value problem
The discrete problem (Q ,h ) reads as: Proof The proof of this lemma is an analogue of that of Lemma 4.1.
Then, we have the error estimate theorem: (4.14) Proof By taking the proof is an analogue of that of Theorem 4.4.
With the analogue of the proof of the Dirichlet case, we have the error estimate in L 2 norm for Neumann case. Theorem 4.8 For u and u ,h are the solutions of (2.4) and (4.12), respectively, we have

Mixed boundary value problem
Since the regularity theorem of (Q ) for mixed boundary case is weak, we will not put a discussion on the error estimates of the discrete problem for this case. Also, we could not find any discussion on this issue in [8,12] etc.

An approximation for discrete problems
In the discrete problem, we find the inner-product (∇u ,h , ∇v h ) or 1 and (f , v h ) D (sincef is the zero extension of f from onto D) are not applicable to computation, because we assumed that has a curved boundary . The integral of the elements crossing becomes a problem when doing computation. Thus, we need a proper approximation. One way is to replace the integral in the open triangle K , K ∩ = ∅, of whereˆ is a polygon with vertices which are the points of intersection between and the triangles' edges.ˆ satisfies The approximation problem of (Q ,h ) is denoted as (Q ,h ).

Dirichlet boundary value problem
The problem (1.4) is considered. We assume that f h is some interpolation off , such that ( f h , v h ) D is applicable to computation and has f h −f 0 ≤ Ch holds. For example, suppose f ∈ C 1 ( ); then we can choose f h is the linear interpolation of f on the vertices ν of triangles for every ν ∈ˆ and zero on other vertices. Before giving the estimate of û ,h − u ,h 1,D , we quote a lemma from [13]. For any open triangle K , we denote Then we have the following theorem.
Since 1 \ˆ 1 =ˆ \ andˆ 1 \ 1 = \ˆ , the above equation can be written as We apply Lemma 5.1 to obtain This, together with the Poincaré inequality, implies the desired result.

Neumann boundary value problem
(Q ,h ) reads as: The proof is an analogue of that of Theorem 5.2, with using Lemmas 5.1 and 5.3.
The error estimates are showed in Figs. 5 and 6 with the logarithm (log) to base 10, from which we see that for fixed , L 2 error behaves as Ch (in Fig. 5, log Er log h ≈ 1) and H 1 error behaves as C √ h (in Fig. 6, log Er log h ≈ 1 2 ). But at the same time, they also have lower bounds even if we allow h to become arbitrarily small, since the error estimates are also bounded by , according to (6.1) and (6.2). And we can observe that, for different , L 2 error has the lower bound approximate to C , and H 1 error is bounded by C √ . This confirms our theoretical results.