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When using the Finite Element Method (FEM) to solve Maxwell's partial differential equations (PDEs), the process involves discretizing the domain into smaller finite elements and representing the electromagnetic field variables within each element using polynomial interpolation functions. This discretization allows us to approximate the continuous PDEs with a set of algebraic equations. Let's go through the general steps of how this conversion is done: Domain Discretization The first step is to divide the continuous domain (where Maxwell's equations are defined) into a collection of smaller, non-overlapping finite elements. These elements can be triangles or quadrilaterals in 2D or tetrahedra or hexahedra in 3D. The arrangement of elements forms a mesh that covers the entire domain. Selecting Basis Functions Within each finite element, the electromagnetic field variables (electric and magnetic fields) are approximated using interpolation functions known as basis functions. These basis functions are typically polynomial functions defined over the element. The order of the polynomial defines the accuracy of the approximation. Linear elements use linear basis functions, quadratic elements use quadratic basis functions, and so on. Formulating the Weak Form The next step is to convert the PDEs into their weak form. This is achieved by multiplying each equation with a set of appropriate test functions (also usually polynomials) and integrating over each finite element. The weak form allows for the derivatives in the PDEs to be relaxed, leading to simpler integrals. Applying Galerkin Method In the FEM, the most common approach is to use the Galerkin method, which chooses the same basis functions for both the field variables and the test functions. This simplifies the formulation and 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. Boundary Conditions and Source Terms Boundary conditions 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. Source terms (current densities, charges, etc.) are also included in the appropriate equations. 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 electromagnetic field variables at each node in the mesh. Post-Processing Once the field variables are determined, further calculations can be performed, such as computing electric and magnetic field intensities, flux densities, and other derived quantities of interest. By following these steps, the FEM transforms the continuous Maxwell's PDEs into a set of discrete algebraic equations, enabling the solution of complex electromagnetic problems involving arbitrary geometries and material properties.

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Finite Element Method (FEM) to solve Maxwell’s partial differential equations (PDEs)

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