The discontinuous Petrov–Galerkin method with optimal test functions (DPG method) is an approximation scheme that makes the use of optimal test functions, cf. [1, 6, 8], feasible by considering broken test norms . Optimal test functions are those which maximize discrete inf-sup numbers, and the broken form of test spaces and norms allows for their local calculation or approximation. In this form, the DPG method has been developed by Demkowicz and Gopalakrishnan; see [8, 9].
The DPG method has been designed having in mind problems where standard methods suffer from locking phenomena (of small inf-sup numbers) or, otherwise, require specific stabilization techniques. This is particularly the case with singularly perturbed problems where DPG schemes have made some contributions [11, 7, 2, 3, 17]. Nevertheless, in the current form most of the schemes are not cheap to implement. On the one hand, corresponding formulations have several unknowns as is the case with mixed finite elements. On the other hand, the efficient approximation of optimal test functions for singularly perturbed problems is ongoing research. For these reasons, advanced DPG techniques are best used for specific problems whereas finite elements are hard to beat when solving uniformly well-posed problems. Though, it has to be said, that in the latter cases DPG schemes can also be efficient and are competitive in general; cf. the software package developed by Roberts .
In this paper, we develop a discretization method that combines DPG techniques with standard finite elements. In this way, one can restrict the use of more expensive DPG approximations to regions where they are beneficial. Examples are reaction-advection-diffusion problems with small diffusivity in a reduced area, or transmission problems that couple a singularly perturbed problem with an unperturbed problem. In a previous publication , we have proposed such a combination with boundary elements to solve transmission problems of the Laplacian in the full space, and studied a singularly perturbed case of reaction diffusion in . In this paper, we follow the general framework from . There, the basis is set by a heterogeneous variational formulation consisting of an ultra-weak one in a bounded domain and variational boundary integral equations for the exterior unbounded part. Here, we combine an ultra-weak formulation with a standard variational form. We remark that this approach of combining different variational formulations has been systematically analyzed in . Indeed, it is not essential to use an ultra-weak formulation for the DPG scheme, any well-posed formulation would work. Though, the overall strategy in  is to employ DPG techniques throughout whereas we combine different discretization techniques.
An important aspect of the DPG method is the inherent guaranteed error control by the computed residuals. For early computational experiments with adaptivity based on these residuals, see . In , an a posteriori error analysis is given that includes data approximation errors. When coupling the DPG method with other discretizations, the inherent residuals for the DPG approximation have to be combined with appropriate estimators. For the case of boundary elements, see [14, 13], and for the DPG method dealing with contact conditions, see . In this paper, we do not specifically deal with a posteriori error estimation.
Having set our heterogeneous formulation, we proceed to rewrite it by using the so-called trial-to-test operator (which maps the test space to the ansatz space). This is only done for the ultra-weak formulation. The whole system then transforms into one where spaces on the ansatz and test sides are identical. In this way, our heterogeneous variational formulation fits the Lax–Milgram framework just as in . We prove coercivity under the condition that the trial-to-test operator is weighted by a sufficiently large constant. Then quasi-optimal convergence of a discretized version follows by standard arguments. When proving coercivity we follow steps that are similar to the ones when studying the coupling of DPG with boundary elements. But whereas  analyzes only the Laplacian, here we set up the scheme and prove coercivity for a general second-order equation of reaction-advection-diffusion type. Throughout, we assume that our problem is uniformly well posed, i.e., we do not study variations for singularly perturbed cases as in . Also note that, since coefficients are variable, transmission problems can be treated the same way by selecting the sub-domains accordingly. One only has to move the possibly non-homogeneous jump data to the right-hand side functional.
The remainder of this paper is as follows: In Section 2, we start by formulating the model problem. A heterogeneous variational formulation is given in Section 2.1. There, we also state its well-posedness and coercivity (Theorem 2.2) and briefly mention a simplified case where continuity across the sub-domain interface is incorporated strongly (Corollary 2.4). The corresponding discrete DPG-FEM scheme is presented in Section 2.2. Its quasi-optimal convergence is announced in Theorem 2.5. Most technical details and proofs are given in Section 3. In Section 4, we report on some numerical experiments.
Furthermore, throughout the paper, suprema are taken over sets excluding the null element, and the notation is used to say that with a constant which does not depend on any quantity of interest. Correspondingly, the notation is used.
2 Mathematical Setting and Main Results
Let , , be a bounded, simply connected polygonal/polyhedral Lipschitz domain with boundary , and normal vector on pointing outside of Ω. We consider the following elliptic problem of diffusion-advection-reaction type: Given , find such that
Here, and denote standard Sobolev spaces, the latter with zero trace on . Furthermore, the coefficients , , () are assumed to be uniformly bounded. Furthermore, we require that the symmetric part of is positive definite and uniformly bounded from below, with minimum eigenvalue larger than or equal to , and that in Ω. These conditions imply that the operator A is bounded and coercive on .
2.1 Heterogeneous Variational Formulation
In order to solve (2.1) by a combination of DPG method and finite elements, we formulate the problem in a heterogeneous way, using different variational forms in different parts of the domain. For ease of illustration, we restrict ourselves to two Lipschitz sub-domains , (again of polygonal/polyhedral form, each with one connected component) with boundaries , , as specified in Figure 1. There, also a notation for the boundary pieces is introduced. In particular, Γ denotes the interface between the sub-domains. The picture indicates that both sub-domains touch the boundary of Ω (where the homogeneous Dirichlet condition is imposed), but this is not essential. For instance, one sub-domain, , can be of annular type so that, in this case, and . Other combinations can be analyzed without difficulty, also including Neumann conditions. Nevertheless, since our analysis centers around proving coercivity of bilinear forms, we need positivity of the combined advection-reaction term on a sub-domain that does not touch the Dirichlet boundary.
For , the following condition holds: If , then there exists such that a.e. in .
Standard and broken Sobolev spaces. Essential for the DPG method is to use broken test spaces. Therefore, at this early stage we consider a partitioning of into (regular non-intersecting) finite elements T such that , and with skeleton .
Before describing the variational formulation, we introduce the Sobolev spaces we need. For a domain , we use the standard spaces , , , and . The trace operator acting on will be denoted simply by . Then we define the trace space and its dual space with canonical norms. The duality pairing on is and extends the bilinear form. Correspondingly, is the bilinear form.
We also need consisting of -functions with vanishing trace on (). Vector-valued spaces and functions will be denoted by bold symbols. Connected with we use the product spaces and with corresponding product norms.
Now, related with are the skeleton trace spaces
Here, is the exterior unit normal vector on , and indicates the standard way of defining normal traces of -functions. The notation (resp. ) is to be understood in the sense that is a -piecewise trace of an element of (resp. of ). These trace spaces are equipped with the norms
and analogously for and . For functions , (they are elements of product spaces) and , , we use the duality pairings
Heterogeneous formulation in . In , where the DPG method will be used, we consider an ultra-weak variational formulation. As mentioned before, this is just for illustration as any other formulation of primal, mixed, dual-mixed or strong type can be used and analyzed analogously to our case; cf. [12, Section 2.3].
The ultra-weak formulation requires additional independent unknowns
Then we test the defining relation of with for , and equation (2.1) with . Integrating by parts element-wise and substituting the corresponding terms by , , and , we obtain
Here, and denote the -piecewise divergence and gradient operators, respectively.
In , we use the standard primal formulation
Solving (2.1) in Ω is equivalent to solving (in appropriate spaces) (2.3) and (2.4) with homogeneous Dirichlet condition on and transmission conditions on Γ. These transmission conditions will be imposed in variational form. For the time being, we replace by in (2.4). Here, we slightly abuse the notation of noting that for with ; cf., e.g., [13, Section 2.2].
We formally distinguish between and . Then our preliminary heterogeneous variational formulation consists in finding
such that and
for any with
This formulation can be used to define the combined DPG-FEM discretization, but requires that is compatible across Γ with the finite element mesh in . We therefore replace the continuity constraint by a variational coupling on Γ that is similar to a DG-bilinear form involving jumps and fluxes across element boundaries. To this end, we abbreviate
and define the coupling bilinear form
We will also need the bilinear form for that corresponds to :
The construction of fulfills two objectives. First, for a function with and continuity on Γ, there holds for any . Therefore, this bilinear form simply replaces the duality in (2.1). Second, by selecting , the bilinear form reduces to
The first term on the right-hand side is needed for consistency. The last two terms are the ones that generate coercivity of the bilinear forms and ; cf. Lemma 3.1 below.
The final combined ultra-weak primal formulation of (2.1) then reads
Note that, although in (2.9b) we test with functions from U, only and the restrictions to Γ of the and -components of are being used there. Therefore, (2.9a) represents problem (2.1) restricted to without interface condition on Γ, whereas (2.9b) represents the problem on and relates the Cauchy data and .
For reference, we explicitly specify the strong form of (2.9a):
By following , one can show that (2.9) is equivalent to (2.1), so that, in particular, (2.9) has a unique solution; see also Remark 2.3 below. However, since we will use different strategies for solving (2.9a) and (2.9b), we need a slightly different representation. To this end, we define the trial-to-test operator by
Here, denotes the canonical inner product in V. Note that with the Riesz operator . Since B is defined on without boundary condition along Γ, it has a non-trivial kernel, and so does Θ. Still, is surjective. Therefore, denoting by the scaled trial-to-test operator (for to be chosen) yields the following equivalent formulation: For given , find such that
One of our main results is the following theorem.
Let be sufficiently large. Then the variational formulation (2.11) is well posed, and equivalent to problem (2.1) in the following sense: If solves (2.1), then , with () and defined by (2.2), satisfies and solves (2.11).
Furthermore, the bilinear form is U-coercive, i.e.,
By the assumptions on Ω, f, , , and γ, problem (2.1) is uniquely solvable. Furthermore, by the derivation of (2.11), if solves (2.1), then as specified in the statement solves (2.11). This can be seen by integrating by parts and noting that since ; cf. (2.7).
The coercivity of will be shown in Section 3.1 under the assumption that is large enough. It is also straightforward to show that this bilinear form is bounded on , as is the linear functional L on U. The Lax–Milgram lemma proves the well-posedness of (2.11) for being large enough. ∎
Theorem 2.2 implies the well-posedness of formulation (2.9), and the equivalence of (2.9) with problem (2.1), in the same form as indicated in the theorem. It is also equivalent to formulation (2.11) when is large enough. Indeed, the solution u of (2.1) defines a solution of (2.9). Furthermore, any solution of (2.9) also solves (2.11), which, for large κ, is well posed by Theorem 2.2. The well-posedness of (2.9) and its equivalence with (2.1) and (2.11) thus follow (under the condition of κ being large). Since (2.1) and (2.9) are independent of κ, they are well posed and equivalent without further restriction.
As previously mentioned, the continuity constraint can also be imposed strongly. In this case, the solution space is
and the coupling bilinear form reduces to
The variational formulation becomes the following: For given , find such that
Let be sufficiently large. Then the variational formulation (2.13) is well posed, and equivalent to problem (2.1) in the following sense: If solves (2.1), then , with () and defined by (2.2), satisfies and solves (2.13).
Furthermore, for sufficiently large , the bilinear form is -coercive, i.e.,
2.2 Combined DPG-FEM Discretization
The coupled DPG-FEM method consists in solving (2.11) within finite-dimensional subspaces . The indices h and p indicate that this can be piecewise polynomial, conforming spaces of certain polynomial degrees. Specifically, the components of that belong to the unknowns will be piecewise polynomial with respect to the mesh and its skeleton . On the other hand, the component of that approximates is piecewise polynomial with respect to a mesh in . In the current form, we do not need compatibility of the meshes , along Γ. The discrete scheme then reads: For given , find such that
Note that this formulation includes the use of optimal test functions for the discretization in ; cf. (2.9a) and the corresponding terms in (2.11) with trial-to-test operator . On the other hand, the part of the problem that belongs to is solved by standard finite elements; cf. the corresponding relation (2.9b).
Our second main result is the following theorem.
If is sufficiently large, then the scheme (2.15) is uniquely solvable and converges quasi-optimally, i.e.,
The statement is a direct implication of the U-coercivity of for large κ by Theorem 2.2, the Lax–Milgram lemma, and Cea’s estimate. ∎
We note that also the discrete scheme can be changed to impose strongly the continuity of the approximations of and across Γ. This only requires compatibility of the meshes and along the interface, conforming subspaces , and replacing the bilinear form in (2.15) by the bilinear form ; cf. (2.14). The quasi-optimal error estimate from Theorem 2.5 then holds analogously.
3 Technical Details and Proof of Coercivity
The bilinear forms and satisfy
for all ().
for (). The coercivity property then follows with the positivity of the symmetric part of and by using either the Poincaré–Friedrichs inequality and in (if ) or Assumption 2.1, i.e., in (). ∎
We continue with some properties of the operator B, cf. (2.10), when restricted to the space incorporating homogeneous Dirichlet boundary conditions on the whole of , that is,
The operator is an isomorphism with
bounded independently of the mesh .
This is a particular case of the different variational formulations studied in [5, Example 3.7]. More generally, in , Carstensen, Demkowicz and Gopalakrishnan proved that by “breaking” a continuous variational formulation of a well-posed problem (by introducing broken test spaces) and using canonical trace norms, this does not alter the well-posedness of the formulation. ∎
Let us introduce the trace space with canonical norm. To simplify the presentation of some technical details we will need the following trace operator:
The boundedness of this operator is immediate, and analogous to the case of the Laplacian on a single domain considered in [14, Lemma 4].
The operator is bounded with bound independent of .
In the following, we identify the kernel of B when acting on the full space . Let us recall that A is the operator of our problem (2.1). For given , we define its A-harmonic extension by
The operator is bounded with bound independent of . Moreover, is a right-inverse of , and the image of is the kernel of B, that is, .
These statements can be proved analogously to the case of the Laplacian; cf. [14, Lemmas 11 and 12]. ∎
Of course, we also need continuity of the bilinear forms , , and . This is straightforward to show and has already been used in the initial part of the proof of Theorem 2.2. We only give the statement.
The bilinear forms , , and are bounded with bounds independent of the mesh .
Let be given. We start with the simple estimate
The last identity is due to the well-known relations of the trial-to-test operator Θ,
According to Lemma 3.4, the operator is bounded,
We continue by considering . In particular, there holds
Noting that, cf. (2.7),
an application of Lemma 3.1 gives
Relation (3.1) can also be written as
so that the previous bound becomes
for a sufficiently large constant . This proves the stated coercivity of .
4 Numerical Experiments
In this section, we report on two numerical experiments. In both of them we choose and, starting from a manufactured solution, we compute the right-hand side function f. The solution of the second experiment does not satisfy the homogeneous Dirichlet boundary condition. In this case, we use a standard approach and extend the inhomogeneous Dirichlet datum into the domain and then shift the resulting terms to the right-hand side. As discrete trial space we use
where and are a mesh and its skeleton in and is a mesh in . Throughout, we use meshes and which are compatible on the interface Γ (although this is not necessary in our analysis). In the definition of , the term denotes the space of -piecewise polynomials of degree k, the term denotes the space of piecewise constant functions on , and the term denotes the space of piecewise affine and continuous functions on which vanish on . The space is the space of piecewise affine, globally continuous functions on which vanish on . The trial-to-test operator with being the Riesz operator is approximated using the discrete Riesz operator with a finite dimensional space , which we choose to be
The resulting method is called practical DPG method, and was analyzed in . In that work, it was shown that the additional discretization error of using instead of V does not degrade the convergence order. Throughout, we use and do not encounter difficulties with this choice. Note that if denotes the exact solution of (2.11) and the discrete solution of (2.15) is , then by definition of the norm there holds
where is the nodal interpolant of with . Likewise,
where is the lowest-order Raviart–Thomas interpolant of , i.e.,
for any . Furthermore, we plot the errors
as well as the so-called energy error of the DPG part
Here, N denotes the overall number of degrees of freedom of . Hence, for all of the errors defined above.
4.1 Experiment 1
We choose and and use the exact solution
The remaining parameters in equation (2.1) are chosen as , , and . In Figure 2, we plot the errors versus the degrees of freedom on a double logarithmic scale. As expected, all the errors behave like , which is plotted in black without markers. In Figure 3, we plot the error on the coupling boundary Γ for the case with mesh width .
4.2 Experiment 2
We choose and and use the exact solution
The remaining parameters in equation 2.1 are chosen as , , , and . The exact solution u has a curved layer of moderate width inside Ω; see Figure 4. In Figure 5, we plot the errors versus the degrees of freedom on a double logarithmic scale. Again, as expected, we obtain the convergence order . In Figure 6, we plot the error on the coupling boundary Γ, again for mesh width . Note that the layer of the exact solution cuts through Γ, and this is reflected in Figure 6.
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About the article
Published Online: 2017-09-30
Published in Print: 2018-10-01
Funding Source: Comisión Nacional de Investigación Científica y Tecnológica
Award identifier / Grant number: 1150056
Award identifier / Grant number: 1170672
Award identifier / Grant number: ACT1118
Award identifier / Grant number: PFB-03
We gratefully acknowledge financial support by CONICYT through FONDECYT projects 1150056 and 1170672, Anillo ACT1118 (ANANUM), and BASAL project CMM, Universidad de Chile, Chile.