When applying the Finite Element Method (FEM) to solve a continuous function, such as a partial differential equation (PDE) or any other field defined over a domain, several steps are followed to convert the continuous problem into a discrete problem suitable for numerical solution. Here are the general steps: Problem Formulation: Clearly define the continuous problem you want to solve. This involves specifying the governing PDEs, boundary conditions, and any other relevant information about the problem. Domain Discretization: Divide the continuous domain into a collection of smaller, non-overlapping finite elements. The choice of elements (triangles, quadrilaterals, tetrahedra, hexahedra, etc.) depends on the dimensionality of the problem and the complexity of the geometry. Selecting Basis Functions: Within each finite element, select suitable basis functions to approximate the field variable of interest (e.g., temperature, displacement, pressure). These basis functions should be able to represent the local variation of the field within the element. Polynomial functions are commonly used as basis functions in FEM. Formulating the Weak Form: Convert the strong form of the PDEs into their weak form. This is achieved by multiplying each equation with a set of test functions and integrating over the domain. The weak form allows for the derivatives in the PDEs to be relaxed, resulting in simpler integrals. Applying Galerkin Method: The most common approach in FEM is to use the Galerkin method, where the same basis functions used to approximate the field variable are also used as test functions. This leads to a symmetric and positive definite system of equations, which is computationally advantageous. Assembling the System of Equations: After applying the Galerkin method and integrating over each element, the result is a set of coupled algebraic equations for each node in the mesh. These equations relate the unknown field values at each node to the values at neighboring nodes and to the known boundary conditions and sources. Incorporating Boundary Conditions and Source Terms: Boundary conditions and source terms are essential to fully define the problem. They are incorporated into the system of equations, typically modifying certain entries in the global stiffness matrix or right-hand side vector. Solving the Algebraic System: The final step is to solve the resulting system of algebraic equations. Various numerical techniques, such as direct solvers or iterative methods, can be employed to obtain the solution for the field variable at each node in the mesh. Post-Processing: Once the field variable values are determined, further calculations can be performed, such as computing gradients, fluxes, or other derived quantities of interest. By following these steps, the FEM converts the continuous problem into a set of discrete algebraic equations, which can be efficiently solved using numerical methods to obtain approximate solutions to the original continuous function. The accuracy of the FEM solution depends on the mesh density (element size) and the order of the basis functions used. Higher-order elements and finer meshes generally lead to more accurate results but also increase computational costs.
