Research Article | | Peer-Reviewed

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

Received: 2 February 2024    Accepted: 4 March 2024    Published: 3 June 2024
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Abstract

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, 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. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is O(hp+2) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.

Published in Applied and Computational Mathematics (Volume 13, Issue 3)
DOI 10.11648/j.acm.20241303.12
Page(s) 58-68
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Finite Element Method, Hyperbolic Problems, Triangular Meshes, Basis Function, Discontinuous Galerkin

1. Introduction
The discontinuous Galerkin method was first applied to solve the neutron equation after that it was planned for initial-value problems . Cockburn and Shu 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 , for hyperbolic problems and for diffusion and convection-diffusion problems . 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. proved that DG solutions of one-dimensional linear and nonlinear hyperbolic problems using p-degree polynomial approximations exhibit ansuper 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 assembled a posteriori error estimates that converge to the true error under mesh refinement on unstructured triangular meshes. Adjerid and Massey 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 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 thatand
(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
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
In order to prove (27),
(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)
Now (44) simplifies to
This establishes (38).
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)
Now (44) simplifies to
Which establishes (39).
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