FEM1D_PROJECT
Project Data onto a 1D Finite Element Mesh

FEM1D_PROJECT is a C++ program which computes the projection of sample data onto a given finite element mesh.

This procedure can be used to determine the least squares approximant to data (actually, to the piecewise linear interpolant of that data) or to determine the finite element coefficients appropriate when recomputing a finite element solution from a fine mesh to a coarse mesh.

The sample data is given simply as two files, one containing the points, the other, the corresponding data values.

The user must also supply two files defining the finite element mesh, containing the points, and the pairs of points that form the elements.

The program takes the sample data, projects it onto the finite element mesh, and determines a file of finite element coefficients that represent the projection of the data onto the finite element space.

The current version of the program can only handle finite element meshes which are made of piecewise linear basis functions.

Usage:

fem1d_project sample_prefix fem_prefix

• sample_prefix_nodes.txt, the node coordinates.
• sample_prefix_values.txt, the data values;

• fem_prefix_nodes.txt, the node coordinates.
• fem_prefix_elements.txt, the nodes that make up each element;
• fem_prefix_values.txt, created by the program, and contains the projections of the data values onto the finite element space, that is, these may be regarded as coefficients of finite element functions representing the projections of the data.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

FEM1D_PROJECT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM1D, a data directory which contains examples of 1D FEM files, three text files that describe a 1D finite element model;

FEM1D, a C++ program which applies the finite element method to a 1D linear two point boundary value problem.

FEM1D_ADAPTIVE, a C++ program which applies the finite element method to a 1D linear two point boundary value problem using adaptive refinement to improve the solution.

FEM1D_BVP_LINEAR, a C++ program which applies the finite element method, with piecewise linear elements, to a two point boundary value problem in one spatial dimension.

FEM1D_HEAT_STEADY, a C++ program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM1D_NONLINEAR, a C++ program which applies the finite element method to a 1D nonlinear two point boundary value problem.

FEM1D_PACK, a C++ library which contains utilities for 1D finite element calculations.

FEM1D_PMETHOD, a C++ program which applies the p-method version of the finite element method to a 1D linear two point boundary value problem.

FEM1D_SAMPLE, a C++ program which samples a scalar or vector finite element function of one variable, defined by FEM files, returning interpolated values at the sample points.

FEM2D_PROJECT, a C++ program which projects a function F(X,Y), given as a data, into a given finite element space of piecewise linear triangular elements.

FEM2D_PROJECT_FUNCTION, a C++ program which projects a function F(X,Y), given as a formula, into a given finite element space of piecewise linear triangular elements.

FEM3D_PROJECT, a C++ program which projects a function F(X,Y,Z), given as a data, into a given finite element space of piecewise linear tetrahedral elements.

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54.

Source Code:

• fem1d_project.cpp, the source code.
• fem1d_project.sh, commands to compile the source code.

Examples and Tests:

POWERS starts with sample data for the vector function f(x)=[ 1, x, x^2], on a grid of 11 equally spaced nodes from 0 to 10, and projects this onto a piecewise linear grid of 6 equally spaced nodes from 0 to 10.

• sample_powers_nodes.txt, the sample nodes.
• sample_powers_values.txt, the sample nodal values.
• fem_powers_nodes.txt, the FEM nodes.
• fem_powers_elements.txt, the FEM elements.
• fem_powers_values.txt, the projected FEM nodal values.

List of Routines:

• MAIN is the main program for FEM1D_PROJECT.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FEM1D_APPROXIMATE approximates data at sample points with an FEM function.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT file.
• PIECEWISE_LINEAR_PRODUCT_QUAD: piecewise linear product integral.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_MIN returns the minimum of two R8's.
• R83_NP_FA factors a R83 system without pivoting.
• R83_NP_SL solves a R83 system factored by R83_NP_FA.
• R8MAT_DATA_READ reads the data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_WRITE writes an R8MAT file.
• R8MAT_ZERO_NEW returns a new zeroed R8MAT.
• R8VEC_BRACKET3 finds the interval containing or nearest a given value.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an I4VEC from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.