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In summary, a rectangular higher order edge element is a type of finite element used in numerical analysis to solve partial differential equations. It is defined by four corner nodes and uses higher degree polynomials to improve solution accuracy. Compared to other finite element types, it has a higher order of continuity and polynomial order, making it better at approximating solutions with complex boundary conditions. However, it also has a higher computational cost and can be challenging to implement. Some advantages of using rectangular higher order edge elements include their accuracy in capturing complex solutions and higher convergence rate. Future improvements include more efficient numerical algorithms, advancements in computing technology, and ongoing research in the field.

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Yes, there are higher order rectangular edge elements that can be used in finite element method for solving electromagnetic problems. One such element is the Serendipity element, which is a higher order rectangular element with edges that can be used for vector fields. This element has nodal points at the midpoints of the edges, allowing for higher order interpolation and accurate representation of the vector field. Other higher order rectangular edge elements such as the Lagrange element and the Hermite element can also be used for solving electromagnetic problems. These elements have additional degrees of freedom compared to the first order triangular edge elements, allowing for a more accurate solution. It is important to carefully select the appropriate element for the specific problem being solved in order to achieve accurate results.

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