Abstract
The paper deals with high-order discontinuous Galerkin (DG) method with the approximation order that exceeds 20 and reaches 100 and even 1000 with respect to one-dimensional case. To achieve such a high order solution, the DG method with finite difference method has to be applied. The basis functions of this method are high-order orthogonal Legendre or Chebyshev polynomials. These polynomials are defined in one-dimensional space (1D), but they can be easily adapted to two-dimensional space (2D) by cross products. There are no nodes in the elements and the degrees of freedom are coefficients of linear combination of basis functions. In this sort of analysis the reference elements are needed, so the transformations of the reference element into the real one are needed as well as the transformations connected with the mesh skeleton. Due to orthogonality of the basis functions, the obtained matrices are sparse even for finite elements with more than thousands degrees of freedom. In consequence, the truncation errors are limited and very high-order analysis can be performed. The paper is illustrated with a set of benchmark examples of 1D and 2D for the elliptic problems. The example presents the great effectiveness of the method that can shorten the length of calculation over hundreds times.
Highlights
In scientific literature the discontinuous Galerkin (DG) methods or the finite elements method are called high-order when polynomial order p is greater than 2, but is not greater than 10, e.g. in the DG method [11,23,24,29,31,35,43] and in the finite element method [5,8,16,20,26,27]
The mesh refinement techniques are still one of the mainstream of scientific research since it is used for automatic mesh adaptation, e.g. [10,15,38,40,48,49] This paper reveals that it is very easy to combine very high-order finite elements with low-order elements in conforming and nonconforming meshes, so the hp-adaptivity may be very easy to perform in the discontinuous Galerkin method with finite difference rules (DGFD) method
The structure of this paper enables a description of the DGFD method with orthogonal basis functions
Summary
In scientific literature the discontinuous Galerkin (DG) methods or the finite elements method are called high-order when polynomial order p is greater than 2, but is not greater than 10, e.g. in the DG method [11,23,24,29,31,35,43] and in the finite element method [5,8,16,20,26,27]. This work presents the remedy for high level truncation error which involves the application of orthogonal basis functions in the finite elements. This work presents the example in which the problem is defined on a quarter of annuls domain but is solved on a square domain discretized by four quadrilateral high-ordered finite elements. The application of very high-ordered orthogonal polynomial functions in the DGFD method with the transformation into the reference element in 1D and 2D is presented in this paper. In the case when 2D non-rectangular finite elements are applied in the mesh, the transformation to the reference square finite element is needed The details of this transformation in the DGFD method are presented in Sect.
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