## 35J25 Boundary value problems for second-order elliptic equations

With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng's conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng's conditions depends on the radius of the truncated domain.

Several classes of optimal control of electromagnetic fields are considered. Special emphasis is
laid on a non-standard $H$-based formulation of the equations of electromagnetism in multiply connected conductors. By this technique, the Maxwell equations can be solved with reduced computational complexity. While the magnetic field $H$ in the conductor is obtained from an elliptic equation
with the $\curl \sigma^{-1} \curl$ operator, an elliptic equation with the $\div \mu \nabla$ operator is set up for a potential $\psi$ in the isolator.
Both equations are coupled by appropriate interface conditions. In all problems, the
electrical current is controlled in the conducting domain. Several types of control functions are discussed. In particular, the problem of sparse optimal control is investigated in a package of electrical wires. For all problems, the associated sensitivity
analysis is performed.

We propose transmission conditions of order $1$, $2$ and $3$ approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order $N$. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.

Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.

We investigate optimal elliptic
regularity (within the scale of Sobolev spaces) of anisotropic
div--grad operators in three dimensions at a multi-material vertex on
the Neumann boundary part of a polyhedral spatial domain. The
gradient of a solution to the corresponding elliptic PDE (in a
neighbourhood of the vertex) is integrable to an index greater than
three.

We investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of square-integrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated.

Let $\Upsilon$ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function $\mu$ is piecewise constant on a polyhedral partition of $\Upsilon$. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on $\mu$ and the intersection angles between interfaces and $\partial \Upsilon$ ensuring that the operator $-\nabla \cdot \mu \nabla$ maps the Sobolev space $W^1,q_0(\Upsilon)$ isomorphically onto $W^-1,q(\Upsilon)$ for some $q > 3$.