RaviartThomas

Raviart-Thomas elements are a family of finite element spaces used in the numerical solution of partial differential equations, particularly in the context of vector fields. They were introduced by Pierre Raviart and Jean-Pierre Thomas in 1977. These elements are particularly useful for solving problems involving vector fields, such as the Stokes equations, which describe the flow of viscous fluids.

Raviart-Thomas elements are defined on simplicial meshes, typically triangles in two dimensions and tetrahedra in three dimensions. They are constructed to satisfy the inf-sup condition, which is crucial for the stability of the finite element method when applied to problems involving vector fields. This condition ensures that the discrete spaces for velocity and pressure in the Stokes problem are compatible.

The Raviart-Thomas spaces are polynomial spaces of degree k, denoted as RTk, where k is a non-negative integer. The space RTk consists of vector fields that are polynomials of degree k on each element and are continuous across element boundaries. The continuity condition is enforced by requiring that the normal component of the vector field is continuous across the boundaries.

One of the key properties of Raviart-Thomas elements is their ability to represent the divergence of vector fields accurately. This makes them particularly well-suited for problems where the divergence of the solution is important, such as in the Stokes equations. The Raviart-Thomas spaces are also known for their optimal approximation properties, which contribute to their effectiveness in numerical simulations.

In summary, Raviart-Thomas elements are a powerful tool in the finite element method for solving problems involving vector fields. Their construction ensures stability and accuracy, making them a valuable choice for a wide range of applications in computational fluid dynamics and other areas of engineering and applied mathematics.