In an earlier investigation [ 1 ] it was shown experimentally that the Interior Penalty Method solves problem 3 - 4 successfully if second order edge functions of the first kind are used. Moreover it was shown that first order edge functions fail to converge to the exact solution as the mesh is refined. In this work we intend to give theoretical explanations of these observations and investigate the error that is introduced by the regularization term in 3.
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Note that facets are always triangular while inner faces are convex polygons with up to six nodes and boundary faces can have virtually any polygonal shape cf. Non-conforming overlap of two submeshes. It is easy to check that this condition is satisfied if two sequences of static sub-meshes are moved against each other. Later on we will seek the discrete solution in the piecewise polynomial space cf. The associated norms and semi-norms are. We assume that the exact solution A of 3 - 4 in the sense of distributions is such that. We begin the proof of the a priori error estimate by showing that the exact solution A fulfills equation 8 :.
Thus all inner jump terms drop out,. Note that the last two sums include only boundary faces. Let us apply the identity to the second term of 13 :. The second term on the right-hand side vanishes because A is a solution of the strong formulation 3. Substitute this back into 15 to get. Now use 16 - 17 to bound the sum over all faces,. This theorem tells us that the total error is bounded by the best approximation error w. Then, by the triangle inequality,.
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Thus, Lax-Milgram assures the well-posedness of the discrete problem. In comparison to standard FEM on conforming meshes one order of convergence is lost. In order to prove the above theorem we will make use of two Lemmas to bound the face contributions.
Now using the usual change of variables together with 22 we obtain. For this observe that using 22 ,. Now combining 23 - 25 gives. In order to simplify notation, C denotes in this proof an arbitrary, positive constant that is independent of h. This can be implemented easily by using a hierarchical basis for the edge functions [ 14 ]. Figure 3. The meshes for the two half spheres.
We can see that although the error depends slightly on the angle, it converges to zero in all three formulations as h is decreasing. This is due to the better approximation properties of the edge functions in the inside of the two hemispheres cf. From a theoretical point of view it remains unclear whether this property carries over to the SWIP formulation 8 , cf. But we also observe that for some angles the lower end of the spectrum tends to zero. This agrees with the observations of [ 8 ]. However in this work we are concerned with the curl - curl source problem 3 - 4.
We attempt to solve the linear system of equations using the conjugate gradient CG method [ 2 ]. In [ 16 ] it is shown that the CG method converges for consistent, symmetric positive semi-definite problems and that its rate of convergence is determined by the non-zero eigenvalues. We are only interested in the curl of the solution, i.
The choice of a particular set of basis functions allows us to split the electromagnetic field into components with a specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space-time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations in comparison with commonly used absorbing boundary conditions.
On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system
In our numerical tests, we observe spectral convergence in the L2 norm and a dissipative behavior for which we provide a theoretical explanation. A finite element method for the solution of the time-dependent Maxwell equations in mixed form is presented. The method allows for local hp -refinement in space and in time. To this end, a space—time Galerkin approach is employed. This allows for obtaining a non-dissipative method.
To obtain an efficient implementation, a hierarchical tensor product basis in space and time is proposed. The abstract reads We present and analyse a space-time discontinuous Galerkin method for wave propagation problems.
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Share this:. The abstract reads We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin method based on Trefftz polynomials. The abstract reads This note is concerned with an optimal control problem governed by the relativistic Maxwell-Newton-Lorentz equations, which describes the motion of charges particles in electro-magnetic fields and consists of a hyperbolic PDE system coupled with a nonlinear ODE. Galan del Sastre and L.
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Advances in discontinuous Galerkin Methods and related topics
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