FFP_SPARSE
Finite Element Solution of Poisson's Equation
on a Triangulated Region

FFP_SPARSE is a C++ program, using double precision arithmetic, which applies the finite element method to solve a form of Poisson's equation over an arbitrary triangulated region, using sparse matrix storage and an iterative solver.

Sparse Matrix Modifications:

This program is a revised version of FREE_FEM_POISSON.

FREE_FEM_POISSON used a direct LINPACK/LAPACK banded matrix solver. The storage requirements for the banded solver are less than for a full storage solver, but still grow quickly as the problem size increases.

FFP_SPARSE reduces the storage requirements even further, by using sparse matrix techniques. The storage format chosen is known as DSP or "sparse triplet" format, which essentially simply saves in three vectors A, IA, JA, which record the value, row and column of every nonzero entry. To solve the linear system, the MGMRES iterative solver was applied. With these modifications, FFP_SPARSE can handle problems much larger than those that FREE_FEM_POISSON could. A new issue is that the iterative solver must be monitored to ensure that convergence is proceeding properly, and has reached the desired tolerance.

The Triangulated Region:

The computational region is unknown by the program. The user specifies it by preparing a file containing the coordinates of the nodes, and a file containing the indices of nodes that make up triangles that form a triangulation of the region.

Normally, the user does not type in this information by hand, but has a program fill in the nodes, and perhaps another program that constructs the triangulation. However, in the simplest case, the user might construct a very crude triangulation by hand, and have TRIANGULATION_REFINE refine it to something more reasonable.

For the following ridiculously small example:

        4----5
        |\   |\
        | \  | \
        |  \ |  \
        |   \|   \
        1----2----3
      

         0.0 0.0
         1.0 0.0
         2.0 0.0
         0.0 1.0
         1.0 1.0
      

        1 2 4
        5 4 2
        2 3 5
      

The Poisson Equation:

The program is set up to handle the linear Poisson equation with a right hand side function, and nonhomogeneous Dirichlet boundary conditions. The state variable U(X,Y) is then constrained by:

        - ( Uxx + Uyy ) + K(x,y) * U(x,y) = F(x,y)  in the region
                                   U(x,y) = G(x,y)  on the boundary
      

A fancier version of the program is eventually intended, which will handle a more interesting nonlinear PDE, and include optional Neumann boundary conditions.

User Interface:

To specify the right hand side function F(x,y), the linear coefficient K(x,y) and the boundary condition function G(x,y), the user has to modify a file containing three routines,

• void rhs ( int node_num, double node_xy[], double node_rhs[] ) evaluates the right hand side of function F(x,y) at a list of nodes.
• void k_coef ( int node_num, double node_xy[], double node_k[] ) evaluates the coefficient function K(x,y) at a list of nodes.
• void dirichlet_condition ( int node_num, double node_xy[], double node_g[] ) evaluates the Dirichlet boundary condition G(X,Y) at a list of nodes.

To run the program, the user compiles the user routines, links them with FFP_SPARSE, and runs the executable.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. If there are too many nodes, the plot may be too cluttered to read. For lower values, however, it is a valuable map of what is going on in the geometry.

The program is also able to write out a file containing the solution value at every node. This file may be used to create contour plots of the solution.

Related Data and Programs:

CVT_TRIANGULATION is a FORTRAN90 program which constructs a CVT triangulation for certain regions.

DSP is a sparse matrix format.

FEM is a data directory which contains a description of the data files that can be used to describe a finite element model.

FEM_50 is a MATLAB finite element program in just 50 lines of code.

FEM_50_HEAT is a modified version of FEM_50 suitable for solving the heat equation.

FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a linear triangle ("T3") mesh.

FEM_BASIS_T6_DISPLAY is a MATLAB program which displays a basis function associated with a quadratic triangle ("T6") mesh.

FEM_IO is a C++ library of routines for reading or writing the node, element and data files that define a finite element model.

FEM_SAMPLE is a C++ library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

FEM_TO_TEC is a MATLAB program that can convert an FEM model into a TEC graphics file.

FEM1D is an executable C++ program which applies the finite element method, with piecewise linear basis functions, to a linear two point boundary value problem;

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

FEM1D_NONLINEAR is a C++ program that applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

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

FEM2D_HEAT is a C++ executable program that solves the time dependent heat equation on the unit square.

FEM2D_POISSON is a C++ executable program for solving Poisson's equation on a square, using the finite element method.

FEMPACK is a C++ library of routines for finite element calculations.

FFH_SPARSE is a MATLAB program that solves the time dependent heat equation in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.

FFNS_SPARSE is a MATLAB program that solves the steady Navier Stokes equations in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.

FFP_SPARSE is also available in a FORTRAN90 version and a MATLAB version.

FFS_SPARSE is a MATLAB program that solves the steady Stokes equations in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.

FREE_FEM_HEAT is a C++ program that solves the time dependent heat equation on an arbitrary triangulated region in 2D.

FREE_FEM_NAVIER_STOKES is a C++ program for solving the steady incompressible Navier Stokes equations in an arbitrary triangulated region, using the finite element method.

FREE_FEM_POISSON is a C++ program that solves the steady Poisson equation on a triangulated 2D region.

FREE_FEM_STOKES is a C++ program that solves the steady Stokes flow equations on a triangulated 2D region.

HOT_PIPE is a sample problem that can be run with FEM_50_HEAT.

HOT_POINT is a sample problem that can be run with FEM_50_HEAT.

MGMRES is a C++ library of routines which applies the restarted GMRES algorithm to solve a sparse linear system.

PLTMG_SINGLE is a FORTRAN77 library of routines for solving elliptic partial differential equations using the finite element method with piecewise linear triangles and the multigrid approach.

PLOT_POINTS is an executable FORTRAN90 program that can make a plot of the nodes that define the region.

TABLE is a file format which is used to store the input and output files used by the program.

TABLE_DELAUNAY is a C++ executable program which constructs a Delaunay triangulation of a region which is described by a set of points.

TABLE_IO is a C++ library of routines which supplies the routines used to read the data files.

TRIANGULATION_ORDER3 is a data directory which contains a description of the format for the two files needed to describe an order 3 triangulation.

TRIANGULATION_ORDER3_CONTOUR is a MATLAB script which can make contour plots of the computed solution.

TRIANGULATION_PLOT is an executable C++ program which can display an image of the triangulation used by the program.

TRIANGULATION_REFINE is an executable C++ program which can refine a triangulation.

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:

• ffp_sparse.C, the source code;
• ffp_sparse.csh, commands to compile the partial program;

Examples and Tests:

• ELL is an L-shaped region.
• LAKE is a simulated lake with an island.

List of Routines:

• MAIN is the main program for FFP_SPARSE.
• ASSEMBLE_POISSON_DSP assembles the system for the Poisson equation.
• AX computes A * X for a sparse matrix.
• BASIS_ONE_T3 evaluates basis functions for a linear triangular element.
• 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.
• DIRICHLET_APPLY_DSP accounts for Dirichlet boundary conditions.
• DSP_IJ_TO_K seeks the compressed index of the (I,J) entry of A.
• DSP_PRINT_SOME prints some of a DSP matrix.
• DTABLE_DATA_READ reads the data from a real TABLE file.
• DTABLE_HEADER_READ reads the header from a real TABLE file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_NAME_SPECIFICATION determines the names of the input files.
• FILE_ROW_COUNT counts the number of row records in a file.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• I4COL_COMPARE compares columns I and J of an I4COL.
• I4COL_SORT_A ascending sorts the columns of an I4COL.
• I4COL_SWAP swaps two columns of an I4COL.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAt, transposed.
• I4VEC2_COMPARE compares pairs of integers stored in two vectors.
• I4VEC2_SORT_A ascending sorts a vector of pairs of integers.
• ITABLE_DATA_READ reads data from an integer TABLE file.
• ITABLE_HEADER_READ reads the header from an integer TABLE file.
• MGMRES applies the restarted GMRES iteration to a linear system.
• MULT_GIVENS applies a Givens rotation to two successive entries of a vector.
• POINTS_PLOT plots a pointset.
• QUAD_RULE sets the quadrature rule for assembly.
• R8_ABS returns the absolute value of an R8.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8VEC_AMAX returns the maximum absolute value in an R8VEC.
• R8VEC_DOT computes the dot product of a pair of R8VEC's.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• REFERENCE_TO_PHYSICAL_T3 maps reference points to physical points.
• 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.
• SOLUTION_EVALUATE evaluates the solution at a point in a triangle.
• SOLUTION_WRITE writes the solution to a file.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• TRIANGLE_AREA_2D computes the area of a triangle in 2D.
• TRIANGULATION_ORDER3_ADJ_COUNT counts adjacencies in a triangulation.
• TRIANGULATION_ORDER3_ADJ_SET2 sets adjacencies in a triangulation.
• TRIANGULATION_ORDER3_BOUNDARY_NODE indicates nodes on the boundary.
• TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle neighbors.
• TRIANGULATION_ORDER3_PLOT plots a triangulation of a set of nodes.

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