Metodi Numerici per Equazioni alle Derivate Parziali

Codes and report for the exam of the Metodi Numerici per Equazioni alle Derivate Parziali¹ course at UniMiB.

Code

Implementation of an "Adaptive 1D 1st-order Lagrange FEM" method for solving the Poisson problem with Dirichlet boundary conditions.

Contents

• /src/* Source code for FEM.
  • /src/builder.m Mesh builder.
  • /src/estimate.m Error estimator.
  • /src/marker.m Mesh marker.
  • /src/refiner.m Mesh refiner.
  • /src/solver.m Poisson problem solver on the mesh.
• report/report.tex Report in LaTeX.
• /tests/* Test source codes for the report.
  • /tests/comparison.m Error comparison between adaptive method and classical refinement method.
  • /tests/comparisonCond.m Comparison of the conditioning number of the stiffness matrix between the adaptive method and the classical refinement method.
  • /tests/condition.m Study of the trend of the conditioning number of matrix A.
  • /tests/errorTrend.m Convergence study of the method on uniform meshes.
  • /tests/graphical.m Qualitative comparison between analytical and numerical solutions.
  • /tests/reliability.m Reliability of the error estimator of the adaptive method.
  • /tests/private/errorEstimate.m Error estimation in H₁ seminorm.

Test Index

Tests used in the course of the report.

• 3: Tests on Uniformly Refined Meshes
  • 3.1: /tests/graphical.m
  • 3.2: /tests/errorTrend.m
  • 3.3: /tests/condition.m
• 4: Adaptive Refinement Method
  • 4.2: /tests/reliability.m
• 5: Comparisons Between the two Approaches
  • 5.1: /tests/comparison.m
  • 5.2: /tests/comparisonCond.m

Additional Notes

The test codes produce result text files and images respectively in ../results/ and ../gallery/ relative to the execution folder.

Footnotes

1. Numerical Methods for Partial Differential Equations. ↩