Finite Element Method Introduction

PDEs · Axiom Academy

LESSON Finite Element Method Introduction A comprehensive introduction to FEM: transforming PDEs into solvable systems through weak formulations, basis functions, and mesh discretization The foundation of FEM is converting a strong form PDE into a weak form . Consider Poisson's equation: The strong form requires the solution to be twice-differentiable. The weak form relaxes this by multiplying by a test function v and integrating by parts: This is the variational formulation . We seek u in a function space such that this equation holds for all test functions v. The weak form only requires first derivatives, making it easier to approximate numerically. 2. Basis Functions and Test Functions We approximate the solution u(x) as a linear combination of basis functions: The coefficients u_j are unknowns to be determined. Common choices include: Linear elements: Piecewise linear "hat" functions Quadratic elements: Piecewise quadratic polynomials Cubic elements: Higher-order approximations for smooth solutions By choosing the test functions v from the same space (Galerkin method), we obtain a system of algebraic equations: The domain is divided into non-overlapping elements (triangles, quadrilaterals, tetrahedra). For each element e: Local computation: Compute element stiffness matrix K^e and load vector f^e Global assembly: Add contributions to global system matrix K and vector f Boundary conditions: Modify system to enforce Dirichlet/Neumann conditions

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