Automated solution of differential equations by the finite by Anders Logg

By Anders Logg

This ebook is an instructional written by way of researchers and builders in the back of the FEniCS venture and explores a complicated, expressive method of the improvement of mathematical software program. The presentation spans mathematical history, software program layout and using FEniCS in functions. Theoretical features are complemented with laptop code that's to be had as free/open resource software program. The publication starts with a unique introductory instructional for newcomers. Following are chapters partially I addressing primary features of the method of automating the construction of finite point solvers. Chapters partly II deal with the layout and implementation of the FEnicS software program. Chapters partly III current the appliance of FEniCS to quite a lot of purposes, together with fluid movement, reliable mechanics, electromagnetics and geophysics.

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This situation may yield a smaller error u - u_e than what is actually true. More accurate representation of the exact solution is easily achieved by interpolating the formula onto a space defined by higher-order elements, say of third degree: Python code Ve = FunctionSpace(mesh, "Lagrange", degree=3) u_e_Ve = interpolate(u_e, Ve) error = (u - u_e_Ve)**2*dx E = sqrt(assemble(error)) To achieve complete mathematical control of which function space the computations are carried out in, we can explicitly interpolate u too: Python code u_Ve = interpolate(u, Ve) error = (u_Ve - u_e_Ve)**2*dx The square in the expression for error will be expanded and lead to a lot of terms that almost cancel when the error is small, with the potential of introducing significant round-off errors.

The result u_box is a BoxField object that supports “finite difference” indexing and an underlying grid suitable for numpy operations on 2D data. Also 1D and 3D meshes (with linear elements) can be turned into BoxField objects. The ability to access a finite element field in the way one can access a finite difference-type of field is handy in many occasions, including visualization and data analysis. coor[X]. The grid attribute is an instance of class UniformBoxGrid. Many plotting programs can be used to visualize the data in u_box.

Coor[X]. The grid attribute is an instance of class UniformBoxGrid. Many plotting programs can be used to visualize the data in u_box. Matplotlib is now a very popular plotting program in the Python world and could be used to make contour plots of u_box. However, other programs like Gnuplot, VTK, and MATLAB have better support for surface plots at the time of this writing. easyviz, which offers a uniform MATLAB-like syntax as interface to various plotting packages such as Gnuplot, matplotlib, VTK, OpenDX, MATLAB, and others.

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