static void test10()
//****************************************************************************80
//
// Purpose:
//
// TEST10 tests FEM1D_BVP_LINEAR.
//
// Discussion:
//
// We want to compute errors and do convergence rates for the
// following problem:
//
// - uxx + u = x for 0 < x < 1
// u(0) = u(1) = 0
//
// exact = x - sinh(x) / sinh(1)
// exact' = 1 - cosh(x) / sinh(1)
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 July 2015
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Dianne O'Leary,
// Scientific Computing with Case Studies,
// SIAM, 2008,
// ISBN13: 978-0-898716-66-5,
// LC: QA401.O44.
//
{
int e_log;
const int e_log_max = 6;
double h1;
double l2;
double mx;
Console.WriteLine("");
Console.WriteLine("TEST10");
Console.WriteLine(" Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)");
Console.WriteLine(" for 0 < x < 1, with U(0) = U(1) = 0.");
Console.WriteLine(" A(X) = 1.0");
Console.WriteLine(" C(X) = 1.0");
Console.WriteLine(" F(X) = X");
Console.WriteLine(" U(X) = X - SINH(X) / SINH(1)");
Console.WriteLine("");
Console.WriteLine(" log(E) E L2error H1error Maxerror");
Console.WriteLine("");
double[] h_plot = new double[e_log_max + 1];
double[] h1_plot = new double[e_log_max + 1];
double[] l2_plot = new double[e_log_max + 1];
double[] mx_plot = new double[e_log_max + 1];
int[] ne_plot = new int[e_log_max + 1];
for (e_log = 0; e_log <= e_log_max; e_log++)
{
int ne = (int)Math.Pow(2, e_log);
int n = ne + 1;
const double x_lo = 0.0;
const double x_hi = 1.0;
double[] x = typeMethods.r8vec_linspace_new(n, x_lo, x_hi);
double[] u = FEM_1D_BVP.fem1d_bvp_linear(n, FEM_Test_Methods.a10, FEM_Test_Methods.c10, FEM_Test_Methods.f10,
x);
ne_plot[e_log] = ne;
h_plot[e_log] = (x_hi - x_lo) / ne;
l2 = FEM_Error.l2_error_linear(n, x, u, FEM_Test_Methods.exact10);
l2_plot[e_log] = l2;
h1 = FEM_Error.h1s_error_linear(n, x, u, FEM_Test_Methods.exact_ux10);
h1_plot[e_log] = h1;
mx = FEM_Error.max_error_linear(n, x, u, FEM_Test_Methods.exact10);
mx_plot[e_log] = mx;
Console.WriteLine(" " + e_log.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + ne.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + l2.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + h1.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + mx.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
Console.WriteLine("");
Console.WriteLine(" log(E1) E1 / E2 L2rate H1rate Maxrate");
Console.WriteLine("");
for (e_log = 0; e_log < e_log_max; e_log++)
{
int ne1 = ne_plot[e_log];
int ne2 = ne_plot[e_log + 1];
double r = ne2 / (double)ne1;
l2 = l2_plot[e_log] / l2_plot[e_log + 1];
l2 = Math.Log(l2) / Math.Log(r);
h1 = h1_plot[e_log] / h1_plot[e_log + 1];
h1 = Math.Log(h1) / Math.Log(r);
mx = mx_plot[e_log] / mx_plot[e_log + 1];
mx = Math.Log(mx) / Math.Log(r);
Console.WriteLine(" " + e_log.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + ne1.ToString(CultureInfo.InvariantCulture).PadLeft(4) + " /" + ne2.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + l2.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + h1.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + mx.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
//
// Create the data file.
//
string data_filename = "data.txt";
List <string> data_unit = new();
for (e_log = 0; e_log <= e_log_max; e_log++)
{
data_unit.Add(" " + ne_plot[e_log]
+ " " + l2_plot[e_log]
+ " " + h1_plot[e_log]
+ " " + mx_plot[e_log] + "");
}
File.WriteAllLines(data_filename, data_unit);
Console.WriteLine("");
Console.WriteLine(" Created graphics data file \"" + data_filename + "\".");
//
// Plot the L2 error as a function of NE.
//
string command_filename = "commands_l2.txt";
List <string> command_unit = new();
string output_filename = "l2.png";
command_unit.Add("# " + command_filename + "");
command_unit.Add("#");
command_unit.Add("# Usage:");
command_unit.Add("# gnuplot < " + command_filename + "");
command_unit.Add("#");
command_unit.Add("set term png");
command_unit.Add("set output '" + output_filename + "'");
command_unit.Add("set xlabel '<---NE--->'");
command_unit.Add("set ylabel '<---L2(NE)--->'");
command_unit.Add("set title 'L2 error versus number of elements NE'");
command_unit.Add("set logscale xy");
command_unit.Add("set size ratio -1");
command_unit.Add("set grid");
command_unit.Add("set style data lines");
command_unit.Add("plot '" + data_filename + "' using 1:2 with points pt 7 ps 2 lc rgb 'blue',\\");
command_unit.Add(" '" + data_filename + "' using 1:2 lw 3 linecolor rgb 'red'");
File.WriteAllLines(command_filename, command_unit);
Console.WriteLine(" Created graphics command file \"" + command_filename + "\".");
//
// Plot the H1 error as a function of NE.
//
command_filename = "commands_h1.txt";
command_unit.Clear();
output_filename = "h1.png";
command_unit.Add("# " + command_filename + "");
command_unit.Add("#");
command_unit.Add("# Usage:");
command_unit.Add("# gnuplot < " + command_filename + "");
command_unit.Add("#");
command_unit.Add("set term png");
command_unit.Add("set output '" + output_filename + "'");
command_unit.Add("set xlabel '<---NE--->'");
command_unit.Add("set ylabel '<---H1(NE)--->'");
command_unit.Add("set title 'H1 error versus number of elements NE'");
command_unit.Add("set logscale xy");
command_unit.Add("set size ratio -1");
command_unit.Add("set grid");
command_unit.Add("set style data lines");
command_unit.Add("plot '" + data_filename + "' using 1:3 with points pt 7 ps 2 lc rgb 'blue',\\");
command_unit.Add(" '" + data_filename + "' using 1:3 lw 3 linecolor rgb 'red'");
File.WriteAllLines(command_filename, command_unit);
Console.WriteLine(" Created graphics command file \"" + command_filename + "\".");
//
// Plot the MX error as a function of NE.
//
command_filename = "commands_mx.txt";
command_unit.Clear();
output_filename = "mx.png";
command_unit.Add("# " + command_filename + "");
command_unit.Add("#");
command_unit.Add("# Usage:");
command_unit.Add("# gnuplot < " + command_filename + "");
command_unit.Add("#");
command_unit.Add("set term png");
command_unit.Add("set output '" + output_filename + "'");
command_unit.Add("set xlabel '<---NE--->'");
command_unit.Add("set ylabel '<---MX(NE)--->'");
command_unit.Add("set title 'Max error versus number of elements NE'");
command_unit.Add("set logscale xy");
command_unit.Add("set size ratio -1");
command_unit.Add("set grid");
command_unit.Add("set style data lines");
command_unit.Add("plot '" + data_filename + "' using 1:4 with points pt 7 ps 2 lc rgb 'blue',\\");
command_unit.Add(" '" + data_filename + "' using 1:4 lw 3 linecolor rgb 'red'");
File.WriteAllLines(command_filename, command_unit);
Console.WriteLine(" Created graphics command file \"" + command_filename + "\".");
}
static void test09()
//****************************************************************************80
//
// Purpose:
//
// TEST09 carries out test case #9.
//
// Discussion:
//
// Use A9, C9, F9, EXACT9, EXACT_UX9.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 June 2014
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Dianne O'Leary,
// Scientific Computing with Case Studies,
// SIAM, 2008,
// ISBN13: 978-0-898716-66-5,
// LC: QA401.O44.
//
{
int i;
const int n = 11;
Console.WriteLine("");
Console.WriteLine("TEST09");
Console.WriteLine(" Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)");
Console.WriteLine(" for 0 < x < 1, with U(0) = U(1) = 0.");
Console.WriteLine(" A9(X) = 1.0");
Console.WriteLine(" C9(X) = 0.0");
Console.WriteLine(" F9(X) = X * ( X + 3 ) * exp ( X ), X <= 2/3");
Console.WriteLine(" = 2 * exp ( 2/3), 2/3 < X");
Console.WriteLine(" U9(X) = X * ( 1 - X ) * exp ( X ), X <= 2/3");
Console.WriteLine(" = X * ( 1 - X ) * exp ( 2/3 ), 2/3 < X");
Console.WriteLine("");
Console.WriteLine(" Number of nodes = " + n + "");
//
// Geometry definitions.
//
double x_first = 0.0;
double x_last = 1.0;
double[] x = typeMethods.r8vec_linspace_new(n, x_first, x_last);
double[] u = FEM_1D_BVP.fem1d_bvp_linear(n, FEM_Test_Methods.a9, FEM_Test_Methods.c9, FEM_Test_Methods.f9, x);
Console.WriteLine("");
Console.WriteLine(" I X U Uexact Error");
Console.WriteLine("");
for (i = 0; i < n; i++)
{
double uexact = FEM_Test_Methods.exact9(x[i]);
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + x[i].ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + u[i].ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + uexact.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + Math.Abs(u[i] - uexact).ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
double e1 = FEM_Error.l1_error(n, x, u, FEM_Test_Methods.exact9);
double e2 = FEM_Error.l2_error_linear(n, x, u, FEM_Test_Methods.exact9);
double h1s = FEM_Error.h1s_error_linear(n, x, u, FEM_Test_Methods.exact_ux9);
double mx = FEM_Error.max_error_linear(n, x, u, FEM_Test_Methods.exact9);
Console.WriteLine("");
Console.WriteLine(" l1 norm of error = " + e1 + "");
Console.WriteLine(" L2 norm of error = " + e2 + "");
Console.WriteLine(" Seminorm of error = " + h1s + "");
Console.WriteLine(" Max norm of error = " + mx + "");
}
static void test00()
//****************************************************************************80
//
// Purpose:
//
// TEST00 carries out test case #0.
//
// Discussion:
//
// - uxx + u = x for 0 < x < 1
// u(0) = u(1) = 0
//
// exact = x - sinh(x) / sinh(1)
// exact' = 1 - cosh(x) / sinh(1)
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 10 July 2015
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Dianne O'Leary,
// Scientific Computing with Case Studies,
// SIAM, 2008,
// ISBN13: 978-0-898716-66-5,
// LC: QA401.O44.
//
{
int i;
const int n = 11;
Console.WriteLine("");
Console.WriteLine("TEST00");
Console.WriteLine(" Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)");
Console.WriteLine(" for 0 < x < 1, with U(0) = U(1) = 0.");
Console.WriteLine(" A(X) = 1.0");
Console.WriteLine(" C(X) = 1.0");
Console.WriteLine(" F(X) = X");
Console.WriteLine(" U(X) = X - SINH(X) / SINH(1)");
Console.WriteLine("");
Console.WriteLine(" Number of nodes = " + n + "");
//
// Geometry definitions.
//
double x_first = 0.0;
double x_last = 1.0;
double[] x = typeMethods.r8vec_linspace_new(n, x_first, x_last);
double[] u = FEM_1D_BVP.fem1d_bvp_linear(n, FEM_Test_Methods.a00, FEM_Test_Methods.c00, FEM_Test_Methods.f00, x);
Console.WriteLine("");
Console.WriteLine(" I X U Uexact Error");
Console.WriteLine("");
for (i = 0; i < n; i++)
{
double uexact = FEM_Test_Methods.exact00(x[i]);
Console.WriteLine(" " + i.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + x[i].ToString(CultureInfo.InvariantCulture).PadLeft(8)
+ " " + u[i].ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + uexact.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + Math.Abs(u[i] - uexact).ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
}
double e1 = FEM_Error.l1_error(n, x, u, FEM_Test_Methods.exact00);
double e2 = FEM_Error.l2_error_linear(n, x, u, FEM_Test_Methods.exact00);
double h1s = FEM_Error.h1s_error_linear(n, x, u, FEM_Test_Methods.exact_ux00);
double mx = FEM_Error.max_error_linear(n, x, u, FEM_Test_Methods.exact00);
Console.WriteLine("");
Console.WriteLine(" l1 norm of error = " + e1 + "");
Console.WriteLine(" L2 norm of error = " + e2 + "");
Console.WriteLine(" Seminorm of error = " + h1s + "");
Console.WriteLine(" Max norm of error = " + mx + "");
}
static void test07()
//****************************************************************************80
//
// Purpose:
//
// TEST07 does an error analysis.
//
// Discussion:
//
// Use A7, C7, F7, EXACT7, EXACT_UX7.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 09 June 2014
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Eric Becker, Graham Carey, John Oden,
// Finite Elements, An Introduction, Volume I,
// Prentice-Hall, 1981, page 123-124,
// ISBN: 0133170578,
// LC: TA347.F5.B4.
//
{
int i;
Console.WriteLine("");
Console.WriteLine("TEST07");
Console.WriteLine(" Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)");
Console.WriteLine(" for 0 < x < 1, with U(0) = U(1) = 0.");
Console.WriteLine(" Becker/Carey/Oden example");
Console.WriteLine("");
Console.WriteLine(" Compute L2 norm and seminorm of error for various N.");
Console.WriteLine("");
Console.WriteLine(" N l1 error L2 error Seminorm error Maxnorm error");
Console.WriteLine("");
int n = 11;
for (i = 0; i <= 4; i++)
{
//
// Geometry definitions.
//
double x_first = 0.0;
double x_last = 1.0;
double[] x = typeMethods.r8vec_linspace_new(n, x_first, x_last);
double[] u = FEM_1D_BVP.fem1d_bvp_linear(n, FEM_Test_Methods.a7, FEM_Test_Methods.c7, FEM_Test_Methods.f7,
x);
double e1 = FEM_Error.l1_error(n, x, u, FEM_Test_Methods.exact7);
double e2 = FEM_Error.l2_error_linear(n, x, u, FEM_Test_Methods.exact7);
double h1s = FEM_Error.h1s_error_linear(n, x, u, FEM_Test_Methods.exact_ux7);
double mx = FEM_Error.max_error_linear(n, x, u, FEM_Test_Methods.exact7);
Console.WriteLine(" " + n.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + e1.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + e2.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + h1s.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + mx.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
n = 2 * (n - 1) + 1;
}
}
static void test06()
//****************************************************************************80
//
// Purpose:
//
// TEST06 does an error analysis.
//
// Discussion:
//
// Use A6, C6, F6, EXACT6, EXACT_UX6.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 February 2012
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Dianne O'Leary,
// Scientific Computing with Case Studies,
// SIAM, 2008,
// ISBN13: 978-0-898716-66-5,
// LC: QA401.O44.
//
{
int i;
Console.WriteLine("");
Console.WriteLine("TEST06");
Console.WriteLine(" Solve -( A(x) U'(x) )' + C(x) U(x) = F(x)");
Console.WriteLine(" for 0 < x < 1, with U(0) = U(1) = 0.");
Console.WriteLine(" A6(X) = 1.0");
Console.WriteLine(" C6(X) = 0.0");
Console.WriteLine(" F6(X) = pi*pi*sin(pi*X)");
Console.WriteLine(" U6(X) = sin(pi*x)");
Console.WriteLine("");
Console.WriteLine(" Compute L2 norm and seminorm of error for various N.");
Console.WriteLine("");
Console.WriteLine(" N l1 error L2 error Seminorm error Maxnorm error");
Console.WriteLine("");
int n = 11;
for (i = 0; i <= 4; i++)
{
//
// Geometry definitions.
//
double x_first = 0.0;
double x_last = 1.0;
double[] x = typeMethods.r8vec_linspace_new(n, x_first, x_last);
double[] u = FEM_1D_BVP.fem1d_bvp_linear(n, FEM_Test_Methods.a6, FEM_Test_Methods.c6, FEM_Test_Methods.f6,
x);
double e1 = FEM_Error.l1_error(n, x, u, FEM_Test_Methods.exact6);
double e2 = FEM_Error.l2_error_linear(n, x, u, FEM_Test_Methods.exact6);
double h1s = FEM_Error.h1s_error_linear(n, x, u, FEM_Test_Methods.exact_ux6);
double mx = FEM_Error.max_error_linear(n, x, u, FEM_Test_Methods.exact6);
Console.WriteLine(" " + n.ToString(CultureInfo.InvariantCulture).PadLeft(4)
+ " " + e1.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + e2.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + h1s.ToString(CultureInfo.InvariantCulture).PadLeft(14)
+ " " + mx.ToString(CultureInfo.InvariantCulture).PadLeft(14) + "");
n = 2 * (n - 1) + 1;
}
}