Ultraconvergence of the derivative of high‐order finite element method for elliptic problems with constant coefficients

2019 ◽  
Vol 36 (1) ◽  
pp. 173-184
Author(s):  
Wenming He
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elliptic Problems ◽  
High Order ◽  
Constant Coefficients ◽  
Element Method

LOW AND HIGH ORDER FINITE ELEMENT METHOD: EXPERIENCE IN SEISMIC MODELING

1994 ◽  
Vol 02 (04) ◽  
pp. 371-422 ◽  
Author(s):  
E. PADOVANI ◽  
E. PRIOLO ◽  
G. SERIANI
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Computational Efficiency ◽  
High Order ◽  
Cholesky Factorization ◽  
Complex Geometries ◽  
Spectral Elements ◽  
Seismic Modeling ◽  
Static Condensation ◽  
Element Method

The finite element method (FEM) is a numerical technique well suited to solving problems of elastic wave propagation in complex geometries and heterogeneous media. The main advantages are that very irregular grids can be used, free surface boundary conditions can be easily taken into account, a good reconstruction is possible of irregular surface topography, and complex geometries, such as curved, dipping and rough interfaces, intrusions, cusps, and holes can be defined. The main drawbacks of the classical approach are the need for a large amount of memory, low computational efficiency, and the possible appearance of spurious effects. In this paper we describe some experience in improving the computational efficiency of a finite element code based on a global approach, and used for seismic modeling in geophysical oil exploration. Results from the use of different methods and models run on a mini-superworkstation APOLLO DN10000 are reported and compared. With Chebyshev spectral elements, great accuracy can be reached with almost no numerical artifacts. Static condensation of the spectral element's internal nodes dramatically reduces memory requirements and CPU time. Time integration performed with the classical implicit Newmark scheme is very accurate but not very efficient. Due to the high sparsity of the matrices, the use of compressed storage is shown to greatly reduce not only memory requirements but also computing time. The operation which most affects the performance is the matrix-by-vector product; an effective programming of this subroutine for the storage technique used is decisive. The conjugate gradient method preconditioned by incomplete Cholesky factorization provides, in general, a good compromise between efficiency and memory requirements. Spectral elements greatly increase its efficiency, since the number of iterations is reduced. The most efficient and accurate method is a hybrid iterative-direct solution of the linear system arising from the static condensation of high order elements. The size of 2D models that can be handled in a reasonable time on this kind of computer is nowadays hardly sufficient, and significant 3D modeling is completely unfeasible. However the introduction of new FEM algorithms coupled with the use of new computer architectures is encouraging for the future.

Sign in / Sign up

Export Citation Format

Share Document