11.3 Finite element methods and their convergence analysis

Finite element methods are powerful tools for solving partial differential equations numerically. They work by breaking down complex problems into simpler parts, making them easier to solve. This approach is widely used in engineering and physics to model real-world phenomena.

In this section, we'll cover the basics of finite element methods, including how they're formulated and implemented. We'll also look at different types of finite element spaces and explore how these methods converge to accurate solutions.

Finite element principles

Basic concepts and formulation

Top images from around the web for Basic concepts and formulation

• 

  GMD - A mixed finite-element discretisation of the shallow-water equations View original

  Is this image relevant?

• 

  Pixels and Their Neighbors — tutorials 0.0.1 documentation View original

  Is this image relevant?

• 

  GMD - A mixed finite-element discretisation of the shallow-water equations View original

  Is this image relevant?

• 

  Pixels and Their Neighbors — tutorials 0.0.1 documentation View original

  Is this image relevant?

1 of 2

Top images from around the web for Basic concepts and formulation

• 

  GMD - A mixed finite-element discretisation of the shallow-water equations View original

  Is this image relevant?

• 

  Pixels and Their Neighbors — tutorials 0.0.1 documentation View original

  Is this image relevant?

• 

  GMD - A mixed finite-element discretisation of the shallow-water equations View original

  Is this image relevant?

• 

  Pixels and Their Neighbors — tutorials 0.0.1 documentation View original

  Is this image relevant?

1 of 2

• Finite element methods (FEM) are numerical techniques for solving partial differential equations (PDEs) by discretizing the domain into smaller subdomains called finite elements
• The solution is approximated by a linear combination of basis functions defined on each finite element, leading to a system of equations that can be solved for the coefficients
• FEM is based on the weak formulation of the PDE, which involves multiplying the equation by a test function and integrating over the domain, allowing for a more flexible choice of approximation spaces

Main steps in FEM

• Mesh generation: Dividing the domain into finite elements (triangles, quadrilaterals, tetrahedra, etc.)
• Choosing basis functions: Selecting appropriate functions to span the finite element space
• Assembling the system of equations: Discretizing the weak formulation using the chosen basis functions and assembling the global stiffness matrix and load vector
• Applying boundary conditions: Incorporating essential (Dirichlet) and natural (Neumann) boundary conditions into the linear system
• Solving the resulting linear or nonlinear system: Using direct or iterative solvers to obtain the coefficients of the finite element approximation

Finite element spaces

Types of finite element spaces

• Lagrange elements: Continuous piecewise polynomial functions defined on a mesh, constructed using interpolation at nodes
• Hermite elements: Continuous piecewise polynomial functions with continuous derivatives, incorporating derivative information at nodes
• Discontinuous Galerkin elements: Discontinuous piecewise polynomial functions, allowing for greater flexibility in the choice of approximation space

Basis functions

• Basis functions are chosen to span the finite element space and are typically defined locally on each element, with support limited to adjacent elements
• The choice of basis functions depends on the specific PDE, the desired order of accuracy, and the regularity of the solution
• Lagrange basis functions are constructed using interpolation at nodes, while Hermite basis functions also incorporate derivative information at nodes
• Examples of basis functions include linear, quadratic, and cubic polynomials on triangular or quadrilateral elements

Convergence of approximations

A priori error estimates

• A priori error estimates provide bounds on the error between the exact solution and the finite element approximation in terms of the mesh size $h$ and the polynomial degree $p$, typically in the form $||u - u_h||_X ≤ C h^α p^β$, where $X$ is a suitable norm and $C$, $α$, and $β$ are constants
• The rate of convergence depends on the regularity of the solution, the choice of finite element space, and the polynomial degree
• For example, linear Lagrange elements on a quasi-uniform mesh yield an error estimate of $O(h)$ in the $H^1$ norm for a second-order elliptic PDE with a sufficiently smooth solution

A posteriori error estimates and adaptivity

• A posteriori error estimates use information from the computed solution to estimate the error and guide adaptive mesh refinement strategies
• These estimates can be based on residuals, gradient recovery techniques, or hierarchical basis methods
• Adaptive mesh refinement improves the accuracy of the solution by locally refining the mesh in regions with high gradients or error indicators, leading to more efficient use of computational resources

Stability analysis

• Stability analysis ensures that small perturbations in the data lead to small changes in the solution, which is crucial for the well-posedness of the discrete problem
• The discrete problem inherits stability properties from the continuous problem, provided that the finite element spaces satisfy certain conditions (e.g., inf-sup condition for mixed formulations)
• Stability is closely related to the choice of finite element spaces and the regularity of the mesh

Implementing finite element methods

Discretization and assembly

• Implementing FEM involves discretizing the weak formulation of the PDE using the chosen finite element space and basis functions
• Local computations: Computing the element stiffness matrices and load vectors by integrating the weak form over each element using quadrature rules
• Assembly: Assembling the local contributions into a global stiffness matrix and load vector, taking into account the connectivity of the elements
• Sparse matrix storage techniques are used to efficiently store and manipulate the global stiffness matrix

Boundary conditions and solving the linear system

• Essential (Dirichlet) boundary conditions are incorporated into the linear system by modifying the stiffness matrix and load vector, while natural (Neumann) boundary conditions are applied to the load vector
• Direct solvers (LU factorization) or iterative solvers (conjugate gradients, multigrid methods) are used to solve the resulting linear system and obtain the coefficients of the finite element approximation
• Preconditioning techniques can be employed to improve the convergence of iterative solvers, especially for large-scale problems

Efficient implementation techniques

• Quadrature rules (Gaussian quadrature) are used for efficient numerical integration over elements
• Parallelization strategies (domain decomposition, parallel assembly) can be employed to speed up computations for large-scale problems
• Adaptive mesh refinement based on a posteriori error estimates can be used to improve the accuracy of the solution while minimizing computational cost
• High-performance computing techniques (vectorization, GPU acceleration) can further enhance the efficiency of FEM implementations