JKU DISSERTATION TNF

JKU DISSERTATION TNF

A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem. The hp -version of FEM combines local mesh refinement h and local increase of the polynomial order of the approximation space p. The main contribution of this work is a general, unified construction principle for H curl – and H div -conforming finite elements of variable and arbitrary order for various element topologies suitable for unstructured hybrid meshes. Further practical advantages will be discussed by means of the following two issues. An analogous principle is used for the construction of H div -conforming basis functions. A short outline of the construction is as follows. 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:

PhD thesis as PDF file 1. An analogous principle is used for the construction of H div -conforming basis functions. Leszek Demkowicz , University of Texas at Austin. Numerical examples illustrate the robustness and performance of the method. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions, we have to perform an orthogonal projection in each iteration step. 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. 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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Since the desired eigenfunctions belong to the orthogonal complement of the dissertatiob functions, we have to perform an orthogonal projection in each iteration step. A short outline of the construction is as follows.

jku dissertation tnf

Leszek DemkowiczUniversity of Texas at Austin. Numerical examples illustrate the robustness and performance of the method.

jku dissertation tnf

This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations. Further practical advantages will be discussed by means of the following two issues.

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A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell without destroying the global exact sequence. Due to the local exact sequence property this jkk already satisfied for simple splitting strategies. PhD thesis as PDF file 1. 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.

jku dissertation tnf

The hp -version of FEM combines local mesh refinement h and local increase of the polynomial order of the approximation space p. 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.

The main contribution of this work is a general, unified construction principle for H curl – and H div -conforming finite tnt of variable and arbitrary order for various element topologies suitable for unstructured hybrid meshes. The gradient fields of higher-order H 1 -conforming shape functions are H curl -conforming and can be dissertatioj explicitly as shape functions for H curl. 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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In the next step we extend the gradient functions to a hierarchical and conforming basis of the desired rnf space. A challenging topic in computational electromagnetics is the Maxwell eigenvalue problem.

An analogous principle is used for the construction of H div -conforming basis functions. 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: For its solution we use the subspace version of the locally optimal preconditioned gradient method.

Robust Schwarz-type methods for Maxwell’s equations rely on a FE-space dissertaation, which also has to provide a tf splitting of the kernel of the curl-operator.