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This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.

This chapter studies some examples of inverse scattering problems in which the data operator fails to be normal. The impedance boundary condition, the reconstruction from limited-angle data, and the reconstruction of near field data, are used as examples. The chapter begins with the direct scattering problem and recalls results on uniqueness and existence of solutions, then the properties of the far field patterns is presented. For the impedance boundary condition, or the limited data case, or the near field data case the corresponding data-operator fails to be normal. Nevertheless, the characterization by the inf-condition holds. The chapter also proves the characterization of the domain D by the convergence of a Picard series for a combination of the self adjoint parts ¦F+F*¦² and (F-F*)²ⁱ of the data operator F. The characterization is based on a general abstract result from functional analysis.

Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C¹ FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ² for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.

This chapter examines the phase error problem and also shows, via a dispersion analysis, that higher order methods can significantly improve phase accuracy. Once a solution is computed it is desirable to assess the accuracy of the solution to determine how to refine the mesh. The next section of the chapter presents a residual based a posteriori error analysis that shows how both the error in the curl of the solution and the divergence needs to be assessed. The final section concerns absorbing boundary conditions, which are often used in preference to the ‘exact’ techniques in Chapters 10-12 to ease the implementation burden. The standard Silver-Muller condition, infinite elements, and the justly popular Perfectly Matched Layer (PML) of Berenger are discussed.

For the mixed boundary value problem, the scattering domain D consists of several components with different types of boundary conditions. The chapter begins by reviewing results on uniqueness and existence, and proves a factorization of the far field operator. Even in the case where only Dirichlet and Neumann boundary conditions occur on the different components of D (which implies normality of the far field operator), the justification of the original Factorization Method remains an open problem. However, if domains are known a priori which separate the different types of components, then a modified Factorization Method can be constructed and justified. Numerical examples are presented.

From the early 1990s, a large number of alternative formulations of the 3+1 evolution equations have been proposed. There are currently more formulations than there are numerical groups capable of testing them. This chapter discusses a small number of formulations, chosen both because they are a representative sample of the different approaches used, and because they correspond to the formulations used by the majority of numerical evolution codes that exist today. Topics covered include well-posedness, the concept of hyperbolicity, hyperbolicity of the ADM equations, the Bona–Masso and NOR formulations, hyperbolicity of BSSNOK, the Kidder–Scheel–Teukolsky family, higher derivative formulations, the Z4 formulation, boundary conditions, radiative boundary conditions, maximally dissipative boundary conditions, and constraint preserving boundary conditions.

This chapter overviews recent progress in the development of patch dynamics, an essential ingredient of the equation-free framework. In many applications we have a given detailed microscopic numerical simulator that we wish to use over macroscopic scales. Patch dynamics uses only simulations within a number of small regions (surrounding macroscopic grid points) in the space-time domain to approximate a discretization scheme for an unavailable macroscopic equation. The approach was first presented and analyzed for a standard diffusion problem in one space dimension; here, we will discuss subsequent efforts to generalize the approach and extend its analysis. We show how one can modify the definition of the initial and boundary conditions to allow patch dynamics to mimic any finite difference scheme, and we investigate to what extent (and at what computational cost) one can avoid the need for specifically designed patch boundary conditions. One can surround the patches with buffer regions, where one can impose (to some extent) arbitrary boundary conditions. The convergence analysis shows that the required buffer for consistency depends on the coefficients in the macroscopic equation; in general, for advection dominated problems, smaller buffer regions–as compared to those for diffusion-dominated problems–suffice.

When the boundary condition is homogeneous, one may think that there is no data at the boundary and that there is no need of boundary estimates in the maximal estimates. This is reminiscent of the case of weakly dissipative symmetric IBVP, and is compatible with the failure of the K.-L. condition at some elliptic boundary frequencies. This chapter constructs a weakly dissipative symmetrizer under appropriate assumptions. This context is the realm of surface waves of finite energy. A paradigm is the Rayleigh waves in linear elasticity.

The nature of an IBVP for a gas depends on the sign of the normal velocity, and on the sub- and super-sonicity. This chapter proceeds to a classification, describing which boundary conditions are dissipative in the classical sense. It gives an explicit though simple construction of the Lopatinskii determinant. It then provides an explicit form of a dissipative Kreiss' symmetrizer under the uniform K.-L. condition. This construction turns out to be much more elegant than the one made in Chapter 5 for general IBVPs.

This chapter completes the investigation of previous chapters on the Dirichlet and Cauchy problems by applying the techniques to other important problems. It selects two directions, the Neumann boundary conditions and the problems posed on manifolds. Section 11.1 introduces the problem and concepts of weak solution, proves a uniqueness result, and presents examples. Section 11.2 reviews the theory for the existence and uniqueness of weak solutions and limit solutions. Section 11.3 provides proof of better estimates and boundedness of solutions in the case of the PME. Section 11.4 examines the mixed problems and problems posed in exterior space domains. The second main topic of this chapter is the theory of PME and GPME on Riemannian manifolds, which is in Section 11.5.

This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.

This chapter provides an elementary introduction to Initial Boundary Value Problems (IBVP). It introduces the notion of dissipative boundary condition, here in a classical sense. Such IBVPs can be treated with the Hille-Yosida Theorem. Maximal estimates are obtained that will be considered in subsequent chapters. Strict dissipativeness is crucial for obtaining boundary estimates.

This chapter presents a standard variational method based on the electric field for the cavity problem to prepare for the finite element approximation of this problem. Mixed boundary conditions (perfectly conducting on one boundary, impedance on another) are assumed. A suitable solution space is described and the Helmholtz decomposition is used to decompose the problem into a simple scalar elliptic problem and a vector problem posed on a Sobolev space of divergence free fields. This space is shown to have a compact inclusion in the space of square integrable vector fields. After a proof of uniqueness of the solution, the Fredholm alternative is used to prove the existence of a solution to the variational problem, and hence show that this variational formulation is appropriate for discretization.

There are many ways to solve partial differential equations numerically. The most popular methods are: finite differencing, finite elements, and spectral methods. This chapter describes the main ideas behind finite differencing methods, since this is the most commonly used approach in numerical relativity. It focuses on methods for the numerical solution of systems of evolution equations of essentially ‘hyperbolic’ type, and does not deal with the solution of elliptic equations, such as those needed for obtaining initial data.

The Factorization Method is not the only sampling method. This final chapter introduces three alternative methods which belong to this class but, in contrast to the Factorization Method, are based on approximation results of solutions of the Helmholtz equation or Laplace equation by special functions. These approximation results are formulated and proven in Section 7.1. Historically, the Factorization Method is a continuation of the Linear Sampling Method which stems from the Dual Space Method of Colton and Monk developed around 1985. Section 7.2 recalls both of these methods. Recently, an interesting and deeper relationship between the Factorization Method and the Linear Sampling Method has been discovered which explains the good results by the latter method. Then the Singular Sources Method by Potthast is presented and mathematically justified. The final section introduces the Ikehata's Probe Method. In the presentations of the latter methods, special emphasis is given to the relationship to the Factorization Method.

Chapter 7 presents with V. Dolejší presents discontinuous Galerkin methods (DCGMs): violated boundary conditions and continuity of the piecewise polynomials are compensated by additional penalty terms in the discrete weak form. This chapter restricts them to linear and quasilinear equations and systems of order 2. hp-variants of DCGMs and numerical experience with the steady compressible Navier-Stokes equations are added. The proof of stability, nearly identical to Chapter 4 based upon the anticrime transformation, is omitted. The relations between the strong and weak forms yield consistency.

This chapter presents parameterization and geometrical issues. This is also one of key points for an efficient OSD platform. It describes different strategies for shape deformation within and without (level set and CAD-Free) computer aided design data structures during optimization. It discusses mesh deformation and remeshing. It discusses the pros and the cons of injection/suction boundary conditions equivalent to moving geometries when the motion is small. Some strategies to couple mesh adaptation and the shape optimization loop are presented. The aim is to obtain a multi-grid effect and improve convergence.

From the standpoint of quantum field theory, the Casimir effect is related to the vacuum polarization that arises in quantization volumes restricted by boundaries or in spaces with nontrivial topology. Both boundaries and the nontrivial topology of space-time can be considered as classical external conditions, on which background the field quantization should be performed. This chapter presents the basic facts related to the quantization procedure for fields of various spins obeying boundary (or identification) conditions. It starts with the classical wave equations and then considers various boundary conditions. The rest of the chapter is devoted to both the canonical and path-integral field quantization procedures in the presence of boundaries and to different representations for the vacuum energy. Propagators with boundary conditions are also introduced. Although fields of different spin are touched upon, the presentation is primarily devoted to the case of the electromagnetic field in the presence of material boundaries.

This chapter considers the simple but most important configuration of two parallel ideal-metal planes. First, the theory of the scalar and electromagnetic Casimir effects between parallel planes is presented. In comparison with Chapter 2, some basic facts are added concerning the relation between local and global approaches and the polarizations of the electromagnetic field. The radiative corrections to the Casimir force are considered. General analytical formulas for the Casimir free energy, entropy, and pressure at nonzero temperature are presented, as well as the limits of low and high temperature. The agreement between the results obtained and thermodynamics is analyzed. The spinor Casimir effect between planes and the Casimir effect for a wedge are also discussed. At the end of the chapter, the dynamic Casimir effect connected with uniformly moving or oscillating planes is briefly considered.