Most of the classical linear partial differential equations arising in mathematical physics can be written in a common operator form. It has been shown in that this form is an evolutionary problem, given by
where stands for the derivative with respect to time, and are bounded linear operators on some Hilbert space ,
is an unbounded skew-selfadjoint operator onand is a given source term.
We are interested in a unique solution of above equation. The theory presented in  deals vanishing initial conditions at , but any non-zero initial condition can be incorporated as data into . For this purpose let and define the weighted -function space
The space is a Hilbert space endowed with the natural inner product given by
for all , where is the inner product of . The associated weighted -function spaces are denoted by for . Now the solution theory of  states: If there exists a and a such that
then for all right hand sides exists a unique solution .
In  this class of problems was investigated numerically using a discontinuous Galerkin approach for the discretisation in time. Here we want to apply a continuous approach, namely the continuous Galerkin-Petrov method [7, 2, 5, 1]. Like in  we will consider a special subclass of the general class of evolutionary problems, although the approach of the time-discretisation is suitable in the general case.
For the precise class of problems let be the spatial dimension and be partitioned into measurable, disjoint sets and . Denoting by
the characteristic function of a domainwe define the linear operators
Additionally let the unbounded skew-selfadjoint operator A be given by
Here denotes the homogeneous Dirichlet boundary conditions. The Hilbert space can now be specified to
The resulting evolutionary problem is now of mixed type. More precisely, on we get an equation of elliptic type, on the equations becomes parabolic while on the problem is hyperbolic.
The solution theory of  does not only give for a final time and the existence of a unique solution . It holds in addition and therefore, by the Sobolev embedding theorem, is continuous in time. If the right-hand side is smoother, say , and and are in the range of , then the continuity in time of the solution is everywhere and we have, see 
In the following we want to utilise this additional regularity. Section 2 gives the precise formulation of the method considered. Section 3 then deals with the existence of discrete solutions, Section 4
presents error estimates and finally Section5 gives some numerical examples and conclusions.
2 Numerical method
Let us start wit a semi-discretisation in time. The semi-discrete variational form of (1.1) uses a decomposition of into disjoint intervals of length for and the piecewise polynomial-in-time spaces
for the trial and the test functions, where is the space of polynomials of degree up to on with values in . Let us localise the scalar product in to the time intervals by
Then the variational formulation using the continuous Galerkin-Petrov method reads:
Find such that for all and
where denotes the weighted -projection into and is the initial value.
In order to compute an approximation to the solution of problem (1.1), we need also to discretise the spatial domain and the function space . Let be discretised into by a regular simplicial mesh that resolves the sets and , and let be the maximal diameter of the cells of . Furthermore, let denote a polynomial degree. Then the discrete spaces are given by
where the spatial spaces are
Here is the space of polynomials of degree up to on the cell and is the Raviart-Thomas-space, defined by
The fully discrete method then reads similarly to (2.1):
Find such that for all and
where and is the initial value.
3 Existence of discrete solution
Let us start by choosing , the -projection of into the test space and denote by
a localised version of .
It holds for all
Let us consider any interval . Then it holds
where the skew-symmetry of and the definition of was used.
For the first term we apply integration by parts and obtain due to the exponential weight
By the -orthogonality it follows
With the general existence assumption and we obtain
Summing over the intervals the statement follows. ∎
With Lemma 3.1 we have control over the -projection of the solution and of the final value of . In order to obtain control of the full solution we consider two approaches.
1) has a trivial null space
In this case we employ the norm equivalence in discrete spaces
with constants independent of and and we obtain
Applying a Young inequality we have
Therefore, in the case of with only trivial null-space the linear system is solvable and we also have continuous dependence of the solution on the data.
2) may have a non-trivial null space
Here we can apply another norm-equivalence to show the solvability of the discrete systems. Using we have the norm equivalence
with constants independent of and . Using a similar approach as before we obtain
With a Young inequality follows
Therefore, also in the general case the linear system is solvable and the solution on each time interval depends continuously on the data and the initial value of the time interval.
4.1 Semi discretisation in time
Note that we do have the Galerkin orthogonality
be the interpolation operator fulfilling locally for alland
We decompose the error into , where
It holds for any
In the following estimates we will bound the terms of interest by a norm of the interpolation error given by
It holds for any
Thus we have control over the weighted -projection of the error and its final value.
In the case of having only a trivial null-space we have immediate control of the full norm due to
for some implying
In the case of having a non-trivial null-space the norm equivalence considered in the existence proof of Section 3 do only provide a bound exponentially increasing in .
4.2 Full discretisation
Let be the maximal time-step length. Similarly to [3, Section 4] we can now proceed and estimate the error of the full discretisation. Here the only difference is the time-discretisation method.
This is a direct consequence of the results in [3, Section 4] combined with the estimates of the previous section. ∎
5 Numerical examples
We consider two examples with unknown solutions. Simulations with known smooth solutions were also made and the theoretical orders were observed. The two following examples show a more realistic behaviour in the case of changing type systems. All computations were done in the finite-element framework , see github.com/SOFE-Developers/SOFE.
5.1 1+1d example
Let us consider as first example one spatial dimension and combine a hyperbolic and an elliptic region. To be more precise, let , , and . As final time we set . The problem is now given by
with homogeneous Dirichlet-conditions for the first component of , the initial condition and a right-hand side , where is the characteristic function of the non-negative time line and
shows plots of the components of the solution in the domain. Note that the first component has a kink along – it is continuous but not differentiable in . As mesh we use an equidistant mesh of cells in and cells in . In order to calculate the errors we use a reference solution instead of the unknown solution . The reference solution is computed on an mesh with polynomial degrees and . Table 1
shows the results for different values of and and polynomial degrees and .
We coupled as the theory gives for smooth the convergence order
if and are proportional. We observe for the continuous Galerkin-Petrov method
in the first column only a convergence rate between 1 and 2. Increasing the polynomial
degree reduces the error, but does not influence the rate much. A reason for this behaviour
could be that is not smooth enough for the error estimates to hold.
For comparison we also computed approximations with the discontinuous Galerkin method from . The errors given in the remaining columns show a similar behaviour with convergence rates between 1 and 2. Nevertheless, the errors are smaller for the discontinuous approach.
5.2 1+2d example
As second example we consider the last example of . Let , , and The problem is given by
shows some snapshots of the first component of the solution , approximated by a numerical simulation. Again we use equidistant meshes with cells in each dimension of space and cells in . As reference solution replacing the unknown exact solution we use an approximation calculated with and , resp. Table 2
shows the results. Similarly to the previous example we do not achieve the optimal convergence order for both methods. And again the discontinuous Galerkin method has smaller errors.
The continuous solution of an evolutionary system with continuous right hand side can be approximated by several methods. Here we investigated the continuous Galerkin-Petrov method, that has optimal convergence order for smooth solutions. We have proved this for the -projection of the error into the test space. For operators
having only a trivial null space it also follows for the full error. The benefit of the continuous method compared to the discontinuous Galerkin method is the continuity that implies a non-dissipative behaviour. In our examples with unknown solutions, that are probably not smooth, the discontinuous Galerkin method is slightly better. Furthermore, these examples show that an increase of the polynomial degree in space over 2 and in time over 1 gives no huge benefit. This is different for smooth solutions – here both methods achieve the theoretical high convergence orders.
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