1. Introduction
The discontinuous Galerkin method was first applied to solve the neutron equation
[11] | Lesaint, P., Raviart, P.: On a finite element method for solving the neutron transport equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic, New York (1974). |
[11]
after that it was planned for initial-value problems
[6] | Krivodonova, L., Flaherty, J. E.: Error estimation for discontinuous Galerkin solutions of twodimensional hyperbolic problems. Adv. Comput. Math. 19, 57–71 (2003). |
[7] | Adjerid, S., Klauser, A.: Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems. J. Sci. Comput. 22, 5–24 (2005). |
[10] | Delfour, M., Hager, W., Trochu, F.: Discontinuous Galerkin methods for ordinary differential equation. Math. Comput. 154, 455–473 (1981). |
[6, 7, 10]
. Cockburn and Shu
[1] | F. Brezzi, L. D. Marini, and E. S ̈uli: ‘‘Discontinuous Galerkin methods for first-order hyperbolic problems.’’ Math. Models Methods Appl. Sci. 14, 2004, 1893–1903. |
[12] | Cockburn, B., Shu, C. W.: TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: general framework. Math. Comput. 52, 411–435 (1989). |
[1, 12]
prolonged the conservation law method to explain first-order hyperbolic partial differential equations. Super convergence properties for DG methods have been planned for ordinary differential equations
[5] | Dubiner, M.: Spectral methods on triangles and other domains. j. Sci. Comput. 6, 345-390(1991). |
[7] | Adjerid, S., Klauser, A.: Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems. J. Sci. Comput. 22, 5–24 (2005). |
[11] | Lesaint, P., Raviart, P.: On a finite element method for solving the neutron transport equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic, New York (1974). |
[5, 7, 11]
, for hyperbolic problems
[3] | Adjerid, S., Massey, T. C.: Superconvergence of discontinuous finite element solution for non-linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 195, 3331-3346 (2006). |
[4] | Adjerid, S., Baccouch, M.: Error analysis for Discontinuous Galerkin Methodsapplied to hyperbolic problems. Part II:a posteriori error estimation (2008, in preparation). |
[6] | Krivodonova, L., Flaherty, J. E.: Error estimation for discontinuous Galerkin solutions of twodimensional hyperbolic problems. Adv. Comput. Math. 19, 57–71 (2003). |
[3, 4, 6]
and for diffusion and convection-diffusion problems
[2] | Adjerid, S., Massey, T. C.: A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 5877-5897 (2002). |
[8] | Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000). |
[9] | Bottcher, K., Rannacher, R.: Adaptive error control in solving ordinary differential equations by the discontinuous Galerkin method. Tech. Report, University of Heidelberg (1996). |
[2, 8, 9]
. DG methods permit discontinuous bases, which simplify both h-refinement and p-refinement. The solution space consists of piecewise continuous polynomial functions relative to a structured or unstructured mesh. As such, it can sharply confinement solution discontinuities relative to the computational mesh. It upholds local conservation on an elemental basis. The DG method has a simple assertion pattern between elements with a common face that makes it useful for parallel computation. Recently, Adjerid et al.
[13] | Adjerid, S., Baccouch, M.: The Discontinuous Galerkin Metod for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis. J Sci Comput (2007) 33: 75–113, https://doi.org/10.1007/s10915-007-9144-x |
[14] | Adjerid, S., Devine, K. D., Flaherty, J. E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002). |
[13, 14]
proved that DG solutions of one-dimensional linear and nonlinear hyperbolic problems using p-degree polynomial approximations exhibit an
super convergence rate at the roots of Radau polynomial of degree
on each element. They furtherproven a strong
superconvergence at the downwind end of every element. Krivodonova and Flaherty
[6] | Krivodonova, L., Flaherty, J. E.: Error estimation for discontinuous Galerkin solutions of twodimensional hyperbolic problems. Adv. Comput. Math. 19, 57–71 (2003). |
[6]
assembled a posteriori error estimates that converge to the true error under mesh refinement on unstructured triangular meshes. Adjerid and Massey
[2] | Adjerid, S., Massey, T. C.: A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 5877-5897 (2002). |
[3] | Adjerid, S., Massey, T. C.: Superconvergence of discontinuous finite element solution for non-linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 195, 3331-3346 (2006). |
[2, 3]
shownsuper convergence results for multi-dimensional problems using rectangular meshes where they showed that the top term in the true local error is spanned by two
degree Radau polynomials in the
and
directions, respectively. They further showed that a p-degree discontinuous finite element solution exhibits
) superconvergence at Radau points obtained as a tensor product of the roots of Radau polynomial of degree
. In this paper, we extend the study of Flaherty and Krivodonova
[6] | Krivodonova, L., Flaherty, J. E.: Error estimation for discontinuous Galerkin solutions of twodimensional hyperbolic problems. Adv. Comput. Math. 19, 57–71 (2003). |
[6]
to show new super convergence results for DG solutions. The triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. This arrangement will be defined more precisely later. The article presents several new
point wisesuper convergence results for the three types of elements and three polynomial spaces. In particular, it shows that the solution on elements of type I is
super convergent at the two vertices of the inflow edge using an appropriate space. Moreover, for some spaces superiorto the space of polynomials of degree p and smaller than the polynomial space of degree p + 1. It exposed additional super convergence points in the interior of each triangle. On elements of type II, the DG solution is
super convergent at the Legendre points on the outflow edge as well as at interior problem-dependent points. On elements of type III, the DG solution is
super convergent at the Legendre points on the outflow edge and for some polynomial spaces the DG solution is
at every point of the outflow edge. This study will extant a super convergence investigation of the local error. These super convergence results still hold on meshes consisting of elements of type III only. In order to hold these super convergence rates for the global solution on general meshes one needs to use estimates of the boundary conditions at the inflow boundary of each element. This is possible on elements whose inflow edges are on the inflow boundary of the domain while on the remaining elements. It accurate the solution by adding an error estimate and use it as an inflow boundary condition.
2. DG Formulation and Preliminary Results
Consider a linear first order hyperbolic scalar problem on a bounded convex polygonal domain
. Let
denote a constant non zero velocity vector. If
denotes the outward unit normal vector, the domain boundary
, where
is the inflow boundary.
is the outflow boundary.
is the characteristic boundary.
Let
denote a smooth function on 𝛺 and consider the following hyperbolic boundary value problem
(1) Subject to the boundary conditions
Where the function
and
are selected such that the exact solution
Let
be real constants.
The domain
is partitioned into a regular mesh having
triangular elements
of diameter
In the remainder of this study, it omit the element index and refer to an arbitrary element by
whenever confusion is unlikely.
Multiply (
1) by a test function
, integrate over an arbitrary element
,
Apply Green’s theorem to write
(2) Where
and
denote the outflow boundary and inflow boundary, respectively, of
. Next we approximate
by a piecewise polynomial function
whose restriction to
is in
consisting of complete polynomial of degree
(3) Here
is a piecewise polynomial not necessarily continuous across inter-element boundaries.
In our error analysis we will also use the following spaces
(4) (5) And
(6) Note that
and
(7) These spaces have the following dimensions dim
dim
and dim
.
Let
denote the space of piecewise polynomial functions
such that the restriction of
to an element
is in
which denotes
,
or
. The discrete DG formulation consists of determining
such that
(8) Where
is the limit from the inflow element sharing
, i.e., if
, then
Next, consider the problem (
8) on an element
such that
Let
be an approximation of the true solution
on
and subtract (
8) from (
2) with
to obtain the DG orthogonality condition for the local error
(9) The map of a physical triangle
having vertices
into the canonical triangle
and
by the standard affine mapping
(10) For simplicity, consider the DG orthogonality on the right angle with vertices
and
which applying the affine mapping (3.10) with
and
leads to
(11) Figure 1. Three types of element.
The triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III
In the analysis, it will use orthogonal polynomials given on the canonical triangle defined by the vertices
and
as
(12) Where
is the Jacobi Polynomial.
(13) And
is the legendre polynomial.
(14) And is complete in the space
. In our analysis we also need the Radua polynomials
(15) The
-degree polynomials
(16) Satisfy the orthogonality condition
(17) And thus provide a basis for
.
Drop the hat and let
and
denote the shifted Jacobi, Legendre and Radua polynomials, respectively, on [0,1]
The finite element spaces
and
are suboptimal, i.e., they contain
-degree terms that do not contribute to global convergence rate, however, they yield
super-convergence rates at some additional interior points which simplifies the a posteriori error estimation procedures described in
[13] | Adjerid, S., Baccouch, M.: The Discontinuous Galerkin Metod for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis. J Sci Comput (2007) 33: 75–113, https://doi.org/10.1007/s10915-007-9144-x |
[13]
If the exact solution is an analytic function, then the local error can be written as a Maclaurin series
(18) Where
Lemma 1: If
satisfies
(19) And for
(20) Then,
(21) Furthermore, Let
and
be the solution of (
1) and (
8), respectively, with
If
such that
is either a triangle, then the local finite error can be written as
(22) Where
(23) Furthermore, at the outflow boundary of the physical element
the local error satisfies
(24) Local error
Now, the first new results for the local error using space
in element
can be stated as:
Theorem 1:
Under the same assumption as in
there exist two constants
and
such that on the outflow edge
(25) (26) Furthermore,
(27) (28) (29) Proof: First note that (
25) is a direct consequence of
(30) Differentiating the auxiliary polynomial
Leads to
(31) Combining this with (
30) yields
(32) The orthogonally condition for
infers that the first term in the right side of (
32) is zero. Now, integrate the second term in (
32) by parts and use
to write
(33) For
and
, apply the orthogonally condition
To establish (
27). The proof of (
28) follows the same line reasoning.
Now, substitute the maclaurin series of the local error in the DGM orthogonally condition on the canonical element (
11) with
and follow the reasoning of the
term to write
(34) Since on the canonical element the outflow edge is the segment
becomes
(35) Testing against
the first in (
35) is zero which establishes (
29).
Equation (
25) infers that the local error is
superconvergent at the roots of Legendre polynomial on the outflow edge.
The following theorem state and prove the same results as for
an interpolant of the exact boundary condition at the roots of
degree Legendre on the inflow edges.
Theorem 2: Under the same assumptions as in
on each inflow boundary edge the properties (
22) and (
25-29) still hold.
Proof: Since the inflow term in the orthogonally condition (11) is not zero in general, substituting the series
(22) in the DG orthogonally condition (
11), using
And collecting terms having the same power of h we obtain (
11)
………………
(36)A direct application reveals that
satisfies
Thus, the
term yields
…………..
(37)From this point on the proof is the same as for theorem 1. Next, consider elements type II and III using the spaces
.
Nothing that
and
, for
, apply the same proof to establish the result of
and
. However, the DG error in the larger spaces
and
satisfies additional orthogonality conditions on elements of type II and III as stated in the next theorem.
Theorem 3: Let
and
. Let
and
be the solution of (1) and (8) respectively with
. If
such that
is either of type II and III, then the local finite element error can be written as in (
25) where the leading term
, satisfies the following conditions
(38) And
(39) If either
i.e.,
is of type III, then the leading term of the local error is zero on the outflow edge
Furthermore, if
,
, then
Similarly, if
,
, then
And
Proof: Inserting the Maclaurin series for the local error (
22) in this DGM orthogonally condition (
35) with
and
the
term leads to (
34) for all
Testing against
we obtain
(40) Which in turn, can be written as
(41) Consider the polynomials
Leads to
(42) Now, integrate the second term in (
41) by parts and using (
42) leads to
(43) Using
, it can be obtained
(44) In order to prove (
39) we set
in (
35) to obtain
(45) This in turn, can be written as
(46) Consider the polynomials
And
(47) Integrate the second term in (
46) by parts and using (
47) leads to
(48) Using
(49) Continue the proof by considering (
37) with
leading to
.
(50)Testing against
, obtain the orthogonality condition on the outflow edge
Since
, it can be established
As a result, (
50) becomes
(51) Testing against
, (3.51) yields
Consider the
degree polynomial
And its derivative
(52) Where, It is used
.
Noting that
write
, where
is a polynomial of degree
. Then
Hence,
(53) Which infers that
is orthogonal to all polynomial in
with respect to the weight function
. Thus,
which completes the proof.
The proof for the case
is similar.
The next theorem is state and prove the super convergence results for elements of type II and III using the space
.
Theorem 5: Under the assumption of theorem 1 and using the polynomial space
the leading term in the local DG error on an element of type II and III satisfies
(54) Moreover, on an element of type III with
(55) And
(56) Similarly, if
.
(57)And
(58) Proof: Inserting the Maclaurin series for the local error (22) in this DGM orthogonally condition (
35) with
and
the
term leads to
On the canonical element the outflow edge is segment
, becomes
(59) Testing against
(60) This can be written as
(61) Where
Differentiating
and
yields
Integrating the second and third term in (61) by parts and using
and
leads to
(61) Using
(62) Note that outflow boundary terms cancel out and (
62) becomes
(63) Thus, it is established (
54).
Corollary 1: Under the statement of theorem 3 on a triangle of type III with
and
is the DG solution, the following results hold.
If
, the superconvergence rates are
of
at the points
(64) Where
are the roots of
in [0,1].
If
we have
superconvergence rates of
at the points
(65) Where
are the roots of Radua polynomial
shifted to
and
are the roots of
shifted to
Finally, if
, the superconvergence rates are having
at every point on the outflow edge
(66) Proof: Note that (
64) and (
66) follows directly from theorem 1 and 3, respectively. In order to prove (
64) it is noted that
Thus, each line
,
For elements of type I consider the DG formulation (8) on the right angle
with vertices
and
mapped to the reference triangle defined by the vertices
and
. The next theorem state and prove new super convergences results for elements of type I using the spaces
.
Theorem 7: Let
and
denote the reference triangle defined by the vertices
and
. Let
and
be a solution of (1) and (8), respectively with
. If
and
, then the local error can be written as (
22) where the leading term
, satisfies the following orthogonality conditions
(67) (68) (69) (70) Furthermore
(71) (72) Finally, the point wisesuper convergence
Proof: The DG orthogonally (
11) can be written as
(73) Let, use
and substituting (
22) in (
73) and collecting the terms with same powers in
, the
term yields
(74) Testing against
establish (
67).
Since the two outflow edges of
are
and
, (74) may be written as
(75) Testing against
the boundary terms in (75) are zero which yields (68)
Next, obtain (69) and (70) from (75) by testing against
and
respectively.
Finally, prove (
71) by considering the polynomial
And
(76) Testing against
in (
75)
And
(77) This can be written as