June 25, 2012.
To solve partial differential equation for complicated problem that cannot be solved with integral equation (IE), particularly for those boundary value problems of fluid mechanics & dynamics, thermal analysis and geophysics and many others, researchers solve these equations with approximation methods: the finite difference (FD) method and the finite element (FE) method. Both methods start by discretizing the model domain into many small meshes. These meshes are rectangles for FD (Fig. 1 (a)) and can be either rectangles, triangles or unstructured for FE (Fig. 1 (b) displaying the triangular meshes for FE). The unknown variables are assigned at nodes which are at the corners of each mesh as displaying in Fig. 1.
Fig.1. (a) Mesh used for the FD method and (b) mesh used for the FE method.
Because of its rectangular mesh, the finite difference (FD) method is therefore easy and straightforward. Calculation time is fast and use less memory requirement. Because of the rectangular mesh, to model complicated case such as, in geophysics, the topography, these advantages become disadvantages. This is because to model the topography, the domain must be subdivided into many small rectangular meshes both in the vertical and horizontal directions. This greatly increases the calculation time and the memory requirement. In the case of with topography or complicated model, the method that is more suitable is the finite element (FE) method. Because meshes do not have to be rectangles, it is therefore easier to adapt to fit the low and high of the topography. This is therefore the advantages of FE. However, if apply FE and FD to the same simple problem with the same number of unknowns, FD will require lesser calculation time than that of FE and also lesser memory than that of FE. Another weakness of FE would be its mathematic is more complicated than that of FD method. All of these are therefore the disadvantages of FE.
Fig. 2. This figure demonstrates how to divides nodes into FD and FE nodes of a hybrid FD FE method. Black nodes for FE nodes and white nodes for FD nodes. FE nodes are applied where there is a slope, while flat area uses FD nodes.
To efficiently solve the problems, it is therefore better to combine the advantages of both methods. Geophysics Research Unit at Mahidol University mixes both FD and FE together by using the most simple and straightforward strategy. Where there is slope, we model it with FE, and where it is flat, we model it with FD as shown in Fig. 2. White nodes are for FD while black nodes are for FE. The equation to be solved will be a mixed of both FE and FD equations. The equation is controlled by coefficients to direct where to calculate with FD and where to calculate with FE.
Fig. 3. (Above) Model used to test the efficiency of the hybrid code. (Below) Calculated data from the hybrid FD FE codes in three different configurations.
Test results with a model shown in Fig. 3 (top) shows that, in the case of similar number of unknown, the solution from FD differs from the exact solutions as much as 10% while only less than 1% for FE method. However, calculation time of FD is a lot faster than that of FE by as much as 50%, and less memory for FD by as much as 25% of FE. Results are as expected. When using a hybrid FD FE to the same model, the accuracy is within 1% similar to that of FE method. At the same time, its calculation and memory requirement increases from those of FD by only less than 5%. This information shows us the efficiency of the hybrid FD FE codes. It is not as difficult as you think!
For more information, please see,
Chatchai Vachiratienchai, SongkhunBoonchaisuk and Weerachai Siripunvaraporn, 2010,
A hybrid finite difference finite element method to incorporate topography for 2D direct current (DC) resistivity modeling,
Physics of the Earth and Planetary Interiors, 183, 426 434.
Assoc. Prof. Dr. Weerachai Siripunvaraporn
Geophysics Research Group; Integrated Physics Center, Thailand Center of Excellence in Physics,
328 Sri Ayuthaya Rd, Bangkok 10400 and
Department of Physics, Faculty of Science,
Mahidol University, 272 Rama 6 Rd., Rachatawee, Bangkok 10400