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1

Numerical solution of the heat advection equation in a two-dimensional domain using the discontinuous Galerkin method

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 3

57--68

EN

This paper presents a numerical solution of the heat advection equation in a two dimensional domain using the Discontinuous Galerkin Method (DGM). The advection equation is widely used in heat transfer problems, particularly in the field of fluid dynamics. The discontinuous Galerkin method is a numerical technique that allows for the solution of partial differential equations using a piecewise polynomial approximation. In this study, DGM is applied to the heat advection equation and its effectiveness in solving the problem is investigated. The findings of this study suggest that the Discontinuous Galerkin Method is a promising approach for solving heat transfer problems in a two-dimensional domain.

2

The comparison of results obtained from the continuous and discontinuous Galerkin Method for the thermoelasticity problem

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 3

77--88

EN

The presented paper is focused on the comparison of the Continuous and Discontinuous Galerkin Methods in terms of thermoelasticity for a cubic element. For this purpose, a numerical model of the phenomenon was built using both methods together with the Finite Element Method (FEM). The comparison of the results of numerical simulation obtained with the use of an original computer program based on the derived final set of FEM equations for both methods is presented.

3

Discontinuous Galerkin method for the three-dimensional problem of thermoelasticity

Węgrzyn-Skrzypczak Ewa

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

115--126

EN

The paper is focused on the mathematical and numerical approaches for the thermoelasticity problem in the three-dimensional domain. The mathematical description of considered problem is based on the second order differential equations of elasticity with the term describing thermal deformations. The numerical model uses the discontinuous Galerkin method which is widely used to solve the problems of hydrodynamics. The presented paper shows the possibility of using the mentioned method to solve the problem of thermomechanics.

4

Application of LDG scheme to solve semi-differential equations

Izadi Mohammad

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 4

27--39

EN

In the current work, we investigate a technique based on discontinuous Galerkin method for the numerical approximation of semi-differential equations with Caputo’s fractional derivative. In this approach, using the natural upwind fluxes enables us to solve the model problem element by element locally in each subintervals and there is no need to solve a full global matrix. Numerical experiments are given to verify the efficiency and accuracy of the proposed method. Numerical solutions are compared with the exact solutions as well as the numerical solutions obtained by other available well-established computational procedures. The results show that the LDG method is more accurate for solving this class of differential equation with relatively low degrees of polynomials and number of elements.

5

The hp nonconforming mesh refinement in discontinuous Galerkin finite element method based on Zienkiewicz-Zhu error estimation

Jaśkowiec J.

Computer Assisted Methods in Engineering and Science

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2016

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Vol. 23, no. 1

43--67

EN

This paper deals with hp-type adaptation in the discontinuous Galerkin (DG) method. The DG method is formulated in this paper with a non-zero mesh skeleton width, which leads to a version of the method called in this paper the interface discontinuous Galerkin (IDG) method. In this formulation, the mesh skeleton has a finite volume and special finite elements are used for discretization. The skeleton spatial calculations are performed using the finite difference or mid-values formulas which are based on the shape functions of the neighbouring finite elements. The Dirichlet boundary conditions are applied using a nonzero width of the material between the outer boundary and a finite element aligned with the boundary. Next, the paper discusses the mesh refinement of hp type. In the IDG method, the mesh does not have to be conforming. The Zienkiewicz-Zhu (ZZ) error indicator is adapted in the IDG method for the purpose of mesh refinement. The paper is illustrated with two-dimensional examples, in which the mesh refinement for an elliptic problem is performed.

6

The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions

Jaśkowiec J.

Computer Assisted Methods in Engineering and Science

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2016

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Vol. 23, no. 2/3

109--132

EN

This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.

7

On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes

Dumbser M., Munz C. D.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

297-310

EN

This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.