A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
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
Given a bounded, open and convex polygonal domain with straight edges, we consider the linear diffusion-reaction problem
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 , may be applied: provided that , for some , find with such that
for all ; for results dealing with finite element approximations of (3), we refer to . 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 .
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
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
i.e., with the notation above, we have , for . We note the continuous Sobolev embedding , i.e.,
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 ,
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
where is Kronecker’s delta. In addition, for , we have
2.3. Trace lifting
Defining the function
i.e., is continuous along the boundary . Similarly, whenever , using (8), we have
There holds the estimate
where is a constant independent of .
There exists a lifting of the boundary data , i.e., in the sense of traces, with
where is a constant independent of .
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
cf. the gray area in Figure 1. The lifting satisfies the boundary condition
Furthermore, we note that
Using (5), we obtain
we see that , and
Employing Lemma 2.1 completes the argument. ∎
2.4. Weak solution
Then, proceeding analogously as in the proof of Lemma 2.1, we deduce that
with a constant independent of and . Consider the regularized problem
where is the boundary function from (9).
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
Thus, the function belongs to . Furthermore, it holds that
as well as
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  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
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., . In addition, referring to [10, Satz 2], cf. also [1, Section 5.1], there holds the a priori error estimate
with a constant independent of the mesh size .
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.
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