9.2 Finite Element Methods

The finite element method is a powerful numerical technique for solving complex partial differential equations. It breaks down problems into smaller, manageable pieces called elements, using basis functions to approximate solutions across these elements.

Implementation involves assembling local element matrices into a global system, which can be solved for various types of PDEs. Accuracy is crucial, with error estimation, convergence analysis, and solution verification playing key roles in ensuring reliable results.

Fundamentals of Finite Element Method

Fundamentals of finite element method

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• Discretization of the domain divides problem into smaller subdomains (elements) with nodes connecting elements
• Basis functions with local support use piecewise polynomial functions to approximate solution
• Solution approximated by linear combination of basis functions with degrees of freedom as coefficients
• Variational formulation develops weak form of PDE through integration by parts
• Matrix formulation yields system of linear equations with stiffness matrix and load vector

Weak forms and Galerkin method

• Weak form derived by multiplying PDE by test function and integrating over domain
• Boundary conditions incorporated as essential (Dirichlet) or natural (Neumann) conditions
• Galerkin method chooses test functions equal to basis functions for discretization
• Resulting linear system $Ax = b$ where $A$ is stiffness matrix and $b$ is load vector

Implementation and Analysis

Assembly of finite element systems

• Local element matrices computed for stiffness using numerical integration (quadrature)
• Local element vectors calculated for load contributions
• Assembly process maps local to global degrees of freedom and sums element contributions
• Sparse matrix storage efficiently stores global system matrix

Implementation for various PDEs

• Elliptic PDEs (Poisson equation) solve steady-state problems with direct solvers
• Parabolic PDEs (heat equation) handle time-dependent problems using schemes (backward Euler, Crank-Nicolson)
• Hyperbolic PDEs (wave equation) address second-order time derivatives with methods (Newmark)
• Boundary conditions incorporated for both essential and natural types

Accuracy of finite element solutions

• Error estimation employs a priori and a posteriori techniques
• Convergence analysis examines order of convergence and mesh refinement strategies
• Stability analysis considers CFL condition for time-dependent problems
• Solution verification compares with analytical solutions and conducts mesh sensitivity studies
• Adaptive mesh refinement uses error indicators for local refinement techniques