We study discontinuous Galerkin methods of order , dG(q), for temporal semi-discretization of the second order hyperbolic problems
where is a self-adjoint, positive definite, uniformly elliptic second-order operator on a Hilbert space .
We may consider, as a prototype equation for such second order hyperbolic equations, with homogeneous Dirichlet boundary conditions. That is, the classical wave equation,
where is a bounded and convex polygonal domain in , , with boundary . We denote and
. The present work applies also to wave phenomena with vector valued solution, such as wave elasticity.
We recall that for the homogeneous parabolic problem
the solution is given by , where the analytic semigroup , , is uniformly bounded. Furthermore, we have the smoothing properties,
that has a crucial impact on analysis of parabolic type problems, in particular, for non-smooth data analysis, . However, such smoothing properties does not hold for the wave equation.
. The discontinuous Galerkin type methods for time and/or space discretization have been studied extensively in the literature for ordinary differential equations and parabolic/hyperbolic partial differential equations; see, for example,[1, 2, 3, 4, 5, 6, 7, 9, 15, 16, 18, 24, 25] and the references therein. In particular, several discontinuous and continuous Galerkin finite element methods, both in time and space variables, for solving second order hyperbolic equations have appeared in the literature, see e.g., [2, 10, 11, 13, 22] and the references therein.
Uniform in time stability analysis, also so-called strong stability or -stability, has been studied for parabolic problems, see, e.g., [8, 16, 24]. An important tool for such analysis is based on the smoothing property (1.3) of the solution operator for parabolic problems, that is due to analytic semigroups.
In , uniform in time stability and error estimates for dG(q), , have been proved using Dunford-Taylor formula based on smoothing properties of the analytic semigroups. For parabolic problems which is perturbed by a memory term, such analysis has been done for dG(0) and dG(1), using the linearity of the basis functions, see . Another way to analyse uniform in time stability is using a lifting operator technique to write the dG(q) formulation in a strong (pointwise) form, see  and the references therein. For the second order hyperbolic problems, say the wave equation, there is no smoothing property for the solution operator, lacking an analytic semigroup.
In the present work, we formulate the discontinuous Galerkin method of arbitrary integer order for (1.1), in particular, for (1.2). We prove energy identity and stability estimates for the discrete problem of a more general form, by considering an extra (artificial) load term in the so called displacement-velocity formulation. This gives the flexibility to obtain optimal order a priori error estimates with minimal regularity requirement on the solution. We prove optimal order a priori error estimates in and norms for the displacement and -norm of the velocity . The stability constants are independent of the length of the time interval, , that means long-time integration without error accumulation is possible. Similar idea has been used for error analysis of the finite element approximation of the second order hyperbolic equations, see, e.g., [14, 23]. Here, we show that how it can be applied for dG(q) time stepping methods, using energy argument, for stability and error analysis. The other contribution of this work is to prove uniform in time optimal order a priori error estimates for dG(0) and dG(1) based on linearity of the approximate solutions, lacking smoothing property for the solution operator. We also think that our techniques for stability and error analysis is straight forward.
The outline of this paper is as follows. We provide some preliminaries and the weak formulation of the model problem, in . In section 3, we formulate the dG(q) method, and we obtain energy identity and stability estimates for the discrete problem of a slightly more general form. Then, in , we prove optimal order a priori error estimates in and norms for the displacement and -norm of the velocity, with minimal regularity requirement on the solution. We also prove uniform in time a priori error estimates for dG(0) and dG(1), based on linearity of the approximate solutions. Finally, numerical experiments are presented in section 5 in order to illustrate the theory.
We let with the inner product and the induced norm . Denote with the energy inner product and the induced norm . Let be defined with homogeneous Dirichlet boundary conditions on , and be the eigenpairs of , i.e.,
It is known that with
and the eigenvectorsform an orthonormal basis for . Then
and we introduce the fractional order spaces
We note that and .
Defining the new variables and , we can write the velocity-displacement form of (1.2) as
for which, the weak form is to find and such that
This equation is used for dG(q) formulation.
3. The discontinuous Galerkin time discretization
Here, we apply the dG method in time variable using piecewise polynomials of degree at most , and we investigate the stability.
3.1. dG(q) formulation
Let be a temporal mesh with time subintervals and steps , and the maximum step-size by . Let and define the finite element space for each space-time ’Slab’ .
We follow the usual convention that a function is left-continuous at each time level and we define , writing
The dG method determines on for by setting , and then
Now, we define the function space consists of functions which are piecewise smooth with respect to the temporal mesh with values in . We note that . Then we define the bilinear form and the linear form by
Then , the solution of discrete problem (3.1), satisfies
We note that the solution of (2.2) also satisfies
These imply the Galerkin’s orthogonality for the error ,
Integration by parts yields an alternative expression for the bilinear form (3.2), as
We note that the framework applies also to spatial finite dimensional function spaces , such as, a continuous Galerkin finite element method of order for discretization in space variable.
In this section we present a stability (energy) identity and stability estimate, that are used in a priori error analysis. In our error analysis we need a stability identity for a slightly more general problem, that is such that
where the linear form is defined by
That is, instead of (2.2), we study stability of the dG(q) discretization of a more general problem
See Remark 4.2.
We define the -projection by
and denote .
Let be a solution of (3.7). Then for any and , we have the energy identity
Moreover, for some (independent of ), we have the stability estimate
We set in (3.7) to obtain
Now writing the first two terms at the left side as
Then, using (for )
We have the identity
4. A priori error estimates
For a given function we define the interpolatant by
where the latter condition is not used for . By standard arguments we then have
where , see .
4.1. Estimates at the nodes
Let and be the solutions of and respectively. Then with and we have
1. We split the error into two terms, recalling the interpolantin (4.1),
Then, using the alternative expression (3.6), we have
Now, by the fact that vanishes at the time nodes and using the definition of , it follows that and are of degree on and hence they are orthogonal to the interpolation error. We conclude that satisfies the equation
That is, satisfies (3.7) with and .
2. Then applying the stability estimate (3.9) and recalling , we have
To prove the first a priori error estimate (4.3), we set . In view of and , we have
We note that (4.5), means that and in (3.7), which is the reason for considering an extra load term in the first equation of (2.2). This way, we can balance between the right operators and suitable norms to get optimal order of convergence with minimal regularity requirement on the solution. Indeed, in , it has been proved that the minimal regularity that is required for optimal order convergence for finite element discretization of the wave equation is one extra derivative compare to the order of convergence, and it cannot be relaxed. This means that the regularity requirement on the solution in our error estimates are minimal.
4.2. Interior estimates
Now, we prove uniform in time a priori error estimates for dG(0) and dG(1), based on the linearity of the basis functions. We define the following norms
Let and and be the solutions of and , respectively. Then there exists a constant such that for ,