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

Muhammad Toriqul Islam; Muhammad Shakhawat Hossain

doi:doi:10.11648/j.acm.20241303.12

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

Muhammad Toriqul Islam, Muhammad Shakhawat Hossain^*

Published in Applied and Computational Mathematics (Volume 13, Issue 3)

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(h^p+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

[11]

after that it was planned for initial-value problems

[6, 7, 10]

. Cockburn and Shu

[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, 7, 11]

, for hyperbolic problems

[3, 4, 6]

and for diffusion and convection-diffusion problems

[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, 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]

assembled a posteriori error estimates that converge to the true error under mesh refinement on unstructured triangular meshes. Adjerid and Massey

[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]

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

. 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

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)

Download: Download full-size image

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

(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]

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

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.

, the superconvergence rates are

at the points

(64)

Where

are the roots of

in [0,1].

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

(78)

Integration by parts

(79)

Noting that the following holds

Prove (79) for

Thus

Combining

(76) and (79), it is obtained

(80)

Which can be written as the originality condition

(81)

Where

Is a polynomial of degree

Thus,

which establishes (71)

The proof of (72) follow the same line of reasoning by considering the polynomial

And

(82)

Testing against

in (82)

(83)

Yields

And

(84)

This can be written as

(85)

By applying integration by parts

(86)

Noting that the following holds

Prove (86) for

Thus

(87)

Combining

(82) and (87)

It can written

Where

Thus,

which establishes (72)

Conclude the proof of this theorem by establishing the point wise super convergence (73) as

Similarly,

leads to

This completes the proof of the theorem.

3. Conclusion

This paper has investigated the error of the numerical solution by applying the Discontinuous Galerkin finite element method for the second-order hyperbolic differential equation. It is a different and straightforward approach to seek error analysis from all other finite element scheme whichis given in the literature. It is 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

superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces.

Abbreviations

DG	Discontinuous Galerkin

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.
[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).
[4]	Adjerid, S., Baccouch, M.: Error analysis for Discontinuous Galerkin Methodsapplied to hyperbolic problems. Part II:a posteriori error estimation (2008, in preparation).
[5]	Dubiner, M.: Spectral methods on triangles and other domains. j. Sci. Comput. 6, 345-390(1991).
[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).
[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).
[10]	Delfour, M., Hager, W., Trochu, F.: Discontinuous Galerkin methods for ordinary differential equation. Math. Comput. 154, 455–473 (1981).
[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).
[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).
[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).

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APA Style

Islam, M. T., Hossain, M. S. (2024). Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Applied and Computational Mathematics, 13(3), 58-68. https://doi.org/10.11648/j.acm.20241303.12

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ACS Style

Islam, M. T.; Hossain, M. S. Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Appl. Comput. Math. 2024, 13(3), 58-68. doi: 10.11648/j.acm.20241303.12

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AMA Style

Islam MT, Hossain MS. Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Appl Comput Math. 2024;13(3):58-68. doi: 10.11648/j.acm.20241303.12

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@article{10.11648/j.acm.20241303.12,
  author = {Muhammad Toriqul Islam and Muhammad Shakhawat Hossain},
  title = {Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {3},
  pages = {58-68},
  doi = {10.11648/j.acm.20241303.12},
  url = {https://doi.org/10.11648/j.acm.20241303.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241303.12},
  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.
},
 year = {2024}
}

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TY  - JOUR
T1  - Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

AU  - Muhammad Toriqul Islam
AU  - Muhammad Shakhawat Hossain
Y1  - 2024/06/03
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AB  - 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.

VL  - 13
IS  - 3
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Author Information

Muhammad Toriqul Islam

Department of Mathematics, University of Barishal, Barishal, Bangladesh
Muhammad Shakhawat Hossain

Department of Mathematics, University of Barishal, Barishal, Bangladesh

Contact Email

http://orcid.org/0000-0001-7509-4652

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Figure 1

Figure 1. Three types of element.

Plain Text BibTeX RIS

APA Style

Islam, M. T., Hossain, M. S. (2024). Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Applied and Computational Mathematics, 13(3), 58-68. https://doi.org/10.11648/j.acm.20241303.12

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ACS Style

Islam, M. T.; Hossain, M. S. Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Appl. Comput. Math. 2024, 13(3), 58-68. doi: 10.11648/j.acm.20241303.12

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AMA Style

Islam MT, Hossain MS. Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method. Appl Comput Math. 2024;13(3):58-68. doi: 10.11648/j.acm.20241303.12

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@article{10.11648/j.acm.20241303.12,
  author = {Muhammad Toriqul Islam and Muhammad Shakhawat Hossain},
  title = {Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {3},
  pages = {58-68},
  doi = {10.11648/j.acm.20241303.12},
  url = {https://doi.org/10.11648/j.acm.20241303.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241303.12},
  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.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

AU  - Muhammad Toriqul Islam
AU  - Muhammad Shakhawat Hossain
Y1  - 2024/06/03
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241303.12
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SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241303.12
AB  - 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.

VL  - 13
IS  - 3
ER  -

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