A Nitsche FEM for Problems with Discontinuous Boundary Conditions

A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions

Ramona Baumann and Thomas P. Wihler Mathematics Institute, University of Bern, Switzerland
Abstract.

We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known singular functions as well as of an -regular part. The latter part is expressed in terms of an elliptic problem with regularized Dirichlet boundary conditions, and can be approximated by means of a Nitsche finite element approach. The discrete solution of the original problem is then defined by adding the singular part of the exact solution to the Nitsche approximation. In this way, the discrete solution can be shown to converge of second order with respect to the mesh size.

Key words and phrases:
Second-order elliptic PDE, discontinuous Dirichlet boundary conditions, Nitsche FEM
2010 Mathematics Subject Classification:
65N30

1. Introduction

Given a bounded, open and convex polygonal domain with straight edges, we consider the linear diffusion-reaction problem

(1)
(2)

where denotes the boundary of , is a nonnegative function, is a source term, and is a possibly discontinuous function on  whose precise regularity will be specified later on.

Various formulations for (1)–(2), where the Dirichlet boundary data does not necessarily belong to , exist in the literature. For instance, the very weak formulation is based on twofold integration by parts of (1) and, thereby, incorporates the Dirichlet boundary conditions in a natural way. It seeks a solution  such that

for any , where we write for the unit outward normal vector to the boundary . Alternatively, the following saddle point formulation, which traces back to the work [9], may be applied: provided that , for some , find  with such that

(3)

for all ; for results dealing with finite element approximations of (3), we refer to [4]. Another related approach is based on weighted Sobolev spaces (accounting for the local singularities of solutions with discontinuous boundary data), and has been analyzed in the context of -type discontinuous Galerkin methods in [7].

The main idea of this paper is to represent the (weak) solution of (1)–(2) in terms of a regular part as well as an explicitly known singular part (Section 2.4). The latter is expressed by means of suitable singular functions which account for the local discontinuities in the Dirichlet boundary data (Section 2.2). Here, it is crucial to ensure that the boundary data of the regular problem is sufficiently smooth as to provide an trace lifting (see Section 2.3). We shall employ a classical Nitsche technique in order to discretize the regular part of the solution, and define the numerical approximation of (1)–(2) by adding back the (exact) singular part (Section 3.2). A numerical experiment (Section 3.3) underlines that our approach provides optimally converging results.

Throughout the paper we shall use the following notation: For an open domain , , and , we denote by  the class of Lebesgue spaces on . For , we write  to signify the -norm on . Furthermore, for an integer , we let be the usual Sobolev space of order  on , with norm  and semi-norm . The set  represents the subspace of  of all functions with zero trace along . If  is represented as a (disjoint) finite union of open sets, that is, , and  is any class of function spaces, then we write  to mean the set of all functions which belong piecewise (with respect to the partition ) to .

2. Problem formulation

The aim of this section is to establish a suitable framework for the weak solution of (1)–(2).

2.1. Notation

Let , with , for , be a finite set of points on the boundary of the polygonal domain , which are numbered in counter-clockwise direction along ; the points in  mark the locations where the Dirichlet boundary condition  from (2) exhibits discontinuities. Furthermore, we denote by , , the open edge which connects the two points  and ; in the sequel, we shall identify indices , , etc.; for instance, we have and , or  and , etc. Moreover, let  signify the interior angle of  at  (in counter-clockwise direction). Finally, for , i.e., , for , we set , and define the one-sided limits

and the jumps , for .

2.2. Singular functions

In the following, based on the partition , we assume that the boundary data from (2) satisfies

(4)

i.e., with the notation above, we have , for . We note the continuous Sobolev embedding , i.e.,

(5)

for a constant . In particular, this implies that the values of and , with  denoting the (edgewise) tangential derivative of  in counter-clockwise direction along , are well-defined. Hence, for , we may consider the singular functions (cf. [8, Lemma 6.1.1]), for ,

(6)

with

Here, denote polar coordinates with respect to a local coordinate system centered at such that on , and on . We note that is harmonic away from , i.e., in . Since  is smooth away from , there holds

(7)

where  is Kronecker’s delta. In addition, for , we have

(8)

for .

2.3. Trace lifting

Defining the function

(9)

with  from (6), and recalling (7), we note that

(10)

i.e., is continuous along the boundary . Similarly, whenever , using (8), we have

(11)
Lemma 2.1.

There holds the estimate

where  is a constant independent of .

Proof.

By definition of , see (9), for any , there holds

Since  is a linear function along both  and  and smooth on , we deduce the bound

where  is a constant depending on  and . Hence,

Using (5), the proof is complete. ∎

The identities (10) and (11) together with the previous lemma imply the following result.

Lemma 2.2.

There exists a lifting of the boundary data , i.e., in the sense of traces, with

(12)

where  is a constant independent of .

Figure 1. Graphical illustration of (local) trace lifting construction.
Proof.

We use a partition of unity approach. Specifically, to each corner  of , we associate a function  such that for any , and , for .

Fix . If , we may assume, without loss of generality, that  coincides with the origin , and the edge  can be placed on the first coordinate axis. Denoting the (Cartesian) coordinates in this system by , we let

see Figure 1 for a graphical illustration. Observe that

where  is the identity function. Then, for , we define the lifting

where

cf. the gray area in Figure 1. The lifting  satisfies the boundary condition

(13)

Furthermore, we note that

Using (5), we obtain

(14)

If , then the function  belongs to , and by the trace theorem, there exists  which again satisfies (13) as well as (14). Therefore, letting

we see that , and

Employing Lemma 2.1 completes the argument. ∎

2.4. Weak solution

Let

(15)

Then, proceeding analogously as in the proof of Lemma 2.1, we deduce that

(16)

with a constant independent of  and . Consider the regularized problem

(17)
(18)

where  is the boundary function from (9).

Proposition 2.3.

Let be a convex and bounded polygonal domain. Then, there exists a unique solution to  (17)–(18) that satisfies the stability bound

(19)

with a constant  depending on , and on .

Proof.

Proposition 2.2 provides the existence of a function with . Since belongs to , elliptic regularity theory in convex polygons (see, e.g., [6, 2, 5]) implies the existence of a unique remainder function with

and

(20)

Thus, the function belongs to . Furthermore, it holds that

as well as

In addition, combining (12) and (20) yields

which, by virtue of (16), results in the bound (19). ∎

Definition 2.4.

We call the function defined by

(21)

with the unique -solution of (17)–(18), the weak solution of (1)–(2).

Remark 2.5.

It can be verified easily that the weak solution defined in (21) belongs to a class of weighted Sobolev spaces; cf., e.g., [2, 3]. The norms of these spaces contain local radial weights at the discontinuity points  of the Dirichlet boundary data, and, thereby, account for possible singularities in the solution of (1)–(2). Based on an inf-sup theory, the work [7] shows that (1)–(2) exhibits a unique solution within this framework.

3. Numerical approximation

The purpose of this section is to discretize (1)–(2) by a finite element approach. Specifically, we will employ a Nitsche method to obtain a numerical approximation of the elliptic problem (17), with the possibly non-homogeneous Dirichlet boundary condition (18). The discrete solution will then be defined similarly as in (21).

3.1. Meshes and spaces

We consider regular, quasi-uniform meshes of mesh size , which partition  into open disjoint triangles and/or parallelograms , i.e., . Each element is an affinely mapped image of the reference triangle  or the reference square , respectively. Moreover, we define the conforming finite element space

where, for , we write to mean either the space  of all polynomials of total degree at most  on  or the space  of all polynomials of degree at most  in each coordinate direction on .

3.2. Nitsche discretization

The classical Nitsche approach [10] for the numerical approximation of (17)–(18) is given by finding  such that

(22)

Here, denoting by  the elementwise gradient operator, we define the bilinear form

as well as the linear functional

with  and  from (9) and (15), respectively. The penalty parameter  appearing in both forms is chosen sufficiently large (but independent of the mesh size) as to guarantee the well-posedness of the weak formulation (22); this can be shown in a similar way as in the context of discontinuous Galerkin methods; see, e.g.,  [1]. In addition, referring to [10, Satz 2], cf. also [1, Section 5.1], there holds the a priori error estimate

(23)

with a constant  independent of the mesh size .

Definition 3.1.

Analogously to (21), we define the discrete solution of (1)–(2) by

(24)

where  is the Nitsche solution from (22), and  are the singular functions from (6).

Theorem 3.2.

Let be the solution of (1)–(2) given by (21), and its discrete counterpart from (24). Then, there holds the a priori error estimate

(25)

with a constant  independent of .

Proof.

We recall the definitions (21) and (24) in order to notice

Therefore, applying (23) yields

Finally, recalling (19) completes the proof. ∎

3.3. Numerical example

On the rectangle we consider the elliptic boundary value problem

with the Dirichlet boundary data  chosen such that the analytical solution is given by

Here, denote polar coordinates in . Note that the solution  is smooth along except at the origin, where it exhibits a discontinuity jump. In particular, it follows that .

Starting from a regular coarse mesh, we investigate the practical performance of the a priori error estimate derived in Theorem 3.2 within a sequence of uniformly refined elements. In Figure 2 we present a comparison of the norm of the error versus the mesh size on a log-log scale for each of the meshes. Our results are in line with the a priori error estimate (25), and show that the discrete solution  from (24) converges of second order with respect to the mesh size . Moreover, in Figure 3 we show the Nitsche solution defined in (22), as well as the computed solution for a mesh consisting of 1024 elements.

Figure 2. error against mesh size compared to a reference line with slope  (expected behaviour).
Figure 3. Nitsche solution (left) and discrete solution (right) based on a uniform mesh with elements.

References

  • [1] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001), 1749–1779.
  • [2] I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order, SIAM J. Math. Anal. 19 (1988), 172–203.
  • [3] I. Babuška and B. Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions, SIAM J. Math. Anal. 20 (1989), no. 4, 763–781.
  • [4] I. Babuška, Error bounds for finite element method, Numer. Math. 16 (1971), no. 4, 322–333.
  • [5] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, no. 1341, Springer-Verlag, 1988.
  • [6] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.
  • [7] P. Houston and T. P. Wihler, Second-order elliptic PDEs with discontinuous boundary data, IMA J. Numer. Anal. 32 (2012), no. 1, 48–74.
  • [8] J. M. Melenk, On generalized finite element methods, Ph.D. thesis, University of Maryland, 1995.
  • [9] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationelle, Ann. Scuola Norm. Sup., Pisa 16 (1962), 305–326.
  • [10] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15.
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