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An analogous principle is used for the construction of H div -conforming basis functions. In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired polynomial space. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence. This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations. By our separate treatment of edge-based, face-based, and cell-based functions, and by including the corresponding gradient functions, we can establish the local exact sequence property:

The main difficulty in the construction of efficient and parameter-robust preconditioners for electromagnetic problems is indicated by the different scaling of solenoidal and irrotational fields in the curl-curl problem. A short outline of the construction is as follows. Leszek Demkowicz , University of Texas at Austin. For its solution we use the subspace version of the locally optimal preconditioned gradient method. By our separate treatment of edge-based, face-based, and cell-based functions, and by including the corresponding gradient functions, we can establish the local exact sequence property:

The main contribution of this work is a general, unified construction principle for H curl – and H div -conforming finite elements of variable xissertation arbitrary order for various element topologies suitable for unstructured hybrid meshes.

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The gradient fields of higher-order H 1 -conforming shape functions are H curl -conforming and can be chosen explicitly as shape functions for H curl. Further practical advantages will be discussed by means of the following two issues.

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Due to the dossertation exact sequence property this is already satisfied for simple splitting strategies. For its solution we use the subspace version of the locally optimal preconditioned gradient method. An analogous principle is used for the construction of H div -conforming basis functions. Considering benchmark problems involving highly singular eigensolutions, we demonstrate the performance of the constructed preconditioners and the eigenvalue solver in combination with hp -discretization on geometrically refined, anisotropic meshes.

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This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations. Robust Schwarz-type methods for Maxwell’s equations rely on a FE-space splitting, which also has to provide a correct splitting of the kernel of the curl-operator.

Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. By our separate treatment of edge-based, face-based, and cell-based functions, and by including the corresponding gradient functions, we can establish the local exact sequence property: The hp -version of FEM combines local mesh refinement h and local increase of the polynomial order of the approximation space p.

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PhD thesis as PDF file 1. In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired polynomial space.

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A short outline of the construction is as follows. The key point is to respect the de Rham Complex already in the construction of the finite element basis functions and not, as usual, only for the definition of the local FE-space.

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The main difficulty in the construction of efficient and parameter-robust preconditioners for electromagnetic problems is indicated by the different scaling of solenoidal and thf fields in the curl-curl problem. Leszek DemkowiczUniversity of Texas at Austin. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence.

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Numerical examples illustrate the robustness and performance of the method. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem.