Continuous Galerkin finite element methods for hyperbolic integro-differential equations
A hyperbolic integro-differential equation is considered, as a model problem, where the convolution kernel is assumed to be either smooth or no worse than weakly singular. Well-posedness of the problem is studied in the context of semigroup of linear operators, and regularity of any order is proved for smooth kernels. Energy method is used to prove optimal order a priori error estimates for the finite element spatial semidiscrete problem. A continuous space-time finite element method of order one is formulated for the problem. Stability of the discrete dual problem is proved, that is used to obtain optimal order a priori estimates via duality arguments. The theory is illustrated by an example.
Key words and phrases:integro-differential equation, linear semigroup theory, continuous Galerkin finite element method, convolution kernel, stability, a priori estimate.
1991 Mathematics Subject Classification:65M60, 45K05
We consider, for any fixed , a hyperbolic type integro-differential equation of the form
(we use ‘’ to denote ‘’) where is a self-adjoint, positive definite, uniformly elliptic second order operator on a Hilbert space. The kernel is considered to be either smooth (exponential), or no worse than weakly singular, and in both cases with the properties that
For our analysis, we define a function by
and, having (1.2), it is easy to see that
Hence, is a completely monotone function, since
and consequently is a positive type kernel, that is, for any and ,
The fractional order kernels, such as Mittag-Leffler type kernels in fractional viscoelasticity, interpolate between smooth (exponential) kernels and weakly singular kernels, that are singular at origin but integrable on finite time intervals , for any , see  and references therein. This is the reason for considering problem (1.1) with convolution kernels satisfying (1.2).
In  well-posedness of a problem, similar to (1.1) with a Mittag-Leffler type kernel, was studied in the framework of the linear semigroup theory. Here we first extend the theory to prove higher regularity of the solution for more smooth kernels, such that a priori error estimates are fulfilled. We prove optimal order a priori error estimate, by energy methods, for finite element spatial semidiscrete approximate solution. This provides an alternative proof to what we presented in , and is straightforward. The continuous space-time finite element method of order one, cG(1)cG(1), is used to formulate the fully dicrete problem. A similar method has been applied to the wave equation in , where adaptive methods based on dual weighted residual (DWR) method has been studied. An energy identity is proved for the discrete dual problem, using the positive type auxiliary function . This is then used to prove and optimal order a priori error estimates by duality. This and , where a posteriori error analysis of this method has been studied via duality, complete the error analysis of this method for model problems similar to (1.1).
The present work also extend previous works, e.g., , , , on quasi-static fractional order viscoelsticity to the dynamic case. Spatial finite element approximation of integro-differential equations similar to (1.1) have been studied in  and , however, for optimal order a priori error estimate for the solution , they require one extra time derivative regularity of the solution. A dynamic model for viscoelasticity based on internal variables is studied in . The memory term generates a growing amount of data that has to be stored and used in each time step. This can be dealt with by introducing “sparse quadrature” in the convolution term . For a different approach based on “convolution quadrature”, see . However, we should note that this is not an issue for exponentially decaying memory kernels, in linear viscoelasticity, that are represented as a Prony series. In this case recurrence relationships can be derived which means recurrence formula are used for history updating, see  and  for more details. In practice, the global regularity needed for a priori error analysis is not present, e.g., due to the mixed boundary conditions, that calls for adaptive methods based on a posteriori error analysis. We plan to address these issues in future work.
In the sequel, in , well-posedness of the problem is proved and high regularity of the solution of the problem with smooth kernels is verified. In , the spatial finite element discretization is studied and, using energy method, optimal order a priori error estimates are proved. The continuous space-time finite element method of order one is applied to the problem in , and stability estimates for the discrete dual problem are obtained. These are then used to prove optimal order a priori error estimates in by duality. Finally, in , we illustrate the theory by a simple example.
2. Well-posedness and regularity
We use the semigroup theory of linear operators to show that there is a unique solution of (1.1), and we prove that under appropriate assumptions on the data we get higher regularity of the solution. In we quote the main framework from , to prove existence and uniqueness, to be complete. Here we restrict to pure homogeneous Dirichlet boundary condition, though the presented framework applies also to mixed homogeneous Dirichlet-Neumann boundary conditions. But it does not admit mixed homogeneuos Dirichlet nonhomogeneous Neumann boundary conditions, and this case has been studied in  for a more general problem, by means of Galerkin approximation method. Then in we extend the semigroup framework to prove regularity of any order for models with smooth kernels. To this end, we specialize to the homogeneous Dirichlet boundary condition.
2.1. Existence and uniqueness
We let , be a bounded convex domain with smooth boundary . In order to describe the spatial regularity of functions, we recall the usual Sobolev spaces with the corresponding norms and inner products, and we denote . We equip with the energy inner product and norm . We recall that is a selfadjoint, positive definite, unbounded linear operator, with , and we use the norms . We note that with mixed homogeneous Dirichlet-Neumann boundary conditions, we have
We extend by for with to be chosen. By adding to both sides of (1.1), changing the variables in the convolution terms and defining , we get
For a given integer number , we use the Taylor expansion of order of the solution at to define the extension for . That is, we set
where we use the notation , with .
Then we write (2.3), together with the initial conditions, as an abstract Cauchy problem and prove well-posedness.
We set and define the Hilbert spaces
We also define the linear operator on such that, for
with domain of definition
Therefore, a solution of (1.1) satisfies the system of delay differential equations, for ,
This can be writen as the abstract Cauchy problem
where and , since
We note that , so that .
We quote from [7, Theorem 2.2], that generates a -semigroup of cotractions on .
The linear operator is an infinitesimal generator of a -semigroup of contractions on the Hilbert space .
Now, we look for a strong solution of the initial value problem (2.4), that is, a function which is differentiable a.e. on with , if , , and a.e. on .
Recalling the assumptions and , we know that if be a strong solution of the abstract Cauchy problem (2.4) with , then is a solution of (1.1) by [7, Lemma 2.1]. Hence, to prove that there is a unique solution for (1.1), we need to prove that there is a unique strong solution for (2.4). This has been proved in [7, Theorem 2.2], if is Lipschitz continuous, using the fact that the linear operator generates a -semigroup of contractions on . Moreover, for some , we have the regularity estimate, for ,
2.2. High order regularity
In order to prove higher regularity of order (), we assume that the bounded domain is convex, and we specialize to the homogeneous Dirichlet boundary condition. Hence, the elliptic regularity estimate holds, that is
with the initial data .
that, with , implies the initial condition
Throughout, obviously any sum is supposed to be suppressed from the formulas, when .
We note that, if we assume , then by Sobolev inequality, and therefore is well-defined.
One can show, by induction and the fact that by (2.6)
we have ,
Now we note that, in (2.7), we have
so that . Therefore, considering continuty of , we have .
Then, in the same way as in the previous section, with , we can reformulate (2.7), with , as the abstract Cauchy problem
where and , since .
In particular, for , we have
with initial data .
Now, we need to show that from a strong solution of the abstract Cauchy problem (2.13), for , we get a solution of the main problem (1.1). Therfore we should prove that the abstract Cauchy problem (2.13) has a unique strong solution, under certain conditions on the data. The proof is by induction, and therefore we recall some facts from , for .
There is a unique solution of (1.1) if with , , and is Lipschitz continuous. Moreover, for some , we have the regularity estimate, for ,
The proof is by induction. The case follows from Theorem 1.
Now, we assume that the lemma valids for some , and we prove that it holds also for . To this end, we show that if be a strong solution of (2.13) (for ) with , then is a strong solution of (2.13) with , that completes the proof by induction assumption.
Since a.e. on , we have, for ,
The first and the third equation implies that satisfies the first order partial differential equation
This, with , has the unique solution , that implies, by integration with respect to ,
From the first and the second equations we obtain equation (2.7) with , that is obtained from equation (2.1) by differentiating . We recall that equations (1.1) and (2.1) are equivalent, that implies equivalence of equations (2.7) and (2.8). Therefore also satisfies (2.8) with . Then, integrating with respect to , we have, for ,
Using these and (2.9) in (2.15) we conclude (2.8), that is equivalent to (2.7). This means that, is a strong solution of (2.13) with . Hence, by induction assumption, is a solution of (1.1), and this completes the proof. ∎
In the next theorem we find the circumstances under which there is a unique strong solution of the abstract Cauchy problem (2.13), that by Lemma 2 implies existence of a unique solution of (1.1) with higher regularity. We also obtain regularity estimates, which are extensions of (2.5) and (2.14).
For a given integer number , let be Lipschitz continuous and with . We also, recalling , assume the following compatibility conditions:
and for ,
Then there is a unique solution of (1.1).
Moreover, for some :
for , we have the regularity estimate
and, for , we have the estimate
1. The case follows from Theorem 1. Then, for a given , we show that
is differentiable almost everywhere on and .
3. Now we prove (ii). By assumption is Lipschitz continuous. Therefore, by a classical result from functional analysis, is differentiable almost everywhere on and , since is a Hilbert space. Then, recalling the assumption and the fact that
we conclude that is differentiable almost everywhere on and , that completes the proof of (ii).
4. Hence, since generates a -semigroup of contractions on by Corollary 1, we conclude, by [11, Corollary 4.2.10], that there exists a unique strong solution for the abstract Cauchy problem (2.13). This, by Lemma 2, proves that there is a unique solution of (1.1), that completes the first part of the theorem.
The unique strong solution of (2.13), is given by
and we recall the fact that , since is an infinitesimal generator of a semigroup of contractions on . Therefore
Since , , and
therefore we have
Hence, considering the assumption that , we have, for some ,
Since, by elliptic regularity estimate (2.6),
so we have
3. The spatial finite elment discretization
The variational form of (1.1) is to find , such that , , and for ,
Let be a convex polygonal domain and be a regular family of triangulations of with corresponding family of finite element spaces , consisting of continuous piecewise polynomials of degree at most , that vanish on (so the mesh is required to fit ). Here is an integer number. We define piecewise constant mesh function for , and for our error analysis we denote . We note that the finite element spaces have the property that
We recall the -projection and the Ritz projection defined by
Then, the spatial finite element discretization of (3.1) is to find such that , and for ,
where and are suitable approximations to be chosen, respectively, for and in .