Support-Operator Finite-Difference Algorithms for General Elliptic Problems

We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u ⋆) with a linear goal functional G and the solution u ⋆ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.

The quantum query complexity of elliptic PDE

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

The title problem is investigated using the differential quadrature method based on the state‐space formalism. The plates, with mixed boundary conditions, may cross over one‐way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed. Copyright © 2006 John Wiley & Sons, Ltd.

We study superradiant scattering for a charged scalar field subject to Robin (mixed) boundary conditions on a charged BTZ black hole background. Scalar field modes having a real frequency do not exhibit superradiant scattering, independent of the boundary conditions applied. For scalar field modes with a complex frequency, no superradiant scattering occurs if the black hole is static. After exploring some regions of the parameter space, we provide evidence for the existence of superradiantly scattered modes with complex frequencies for a charged and rotating BTZ black hole. Most of the superradiantly scattered modes we find satisfy Robin (mixed) boundary conditions, but there are also superradiantly scattered modes with complex frequencies satisfying Dirichlet and Neumann boundary conditions. We explore the effect of the black hole and scalar field charge on the outgoing energy flux of these superradiantly scattered modes, and also investigate their stability.

In this paper, the author introduces the mathematical concepts regarding torsion functions and their derivations under the Robin and Dirichlet boundary conditions which are integral in mechanical engineering, theoretical physics, and other disciplines. Functions that satisfy given or required partial differential equations contain a wealth of information about materials or structures when subjected to stress. We examine these functions under two types of boundary conditions: Robin, which contains Dirichlet and Neumann conditions, Dirichlet that provides conditions on the boundary. Also, in this section, we formally describe the mathematical model used in the study and provide some definitions, notations, and basic propositions that form the foundation for subsequent analysis. In answer to such concerns, the torsion function problems solved herein utilize analytical and numerical computations to reveal the differences and similarities in behavior between the two boundary conditions. Strong empirical evidence further supports our findings, proving the real-life relevance of the revealed theoretical concepts. Lastly, the implications for these findings in the real-world application are also presented and analysed with regards to advantages and disadvantages of various boundary conditions in the engineering of systems. We end this paper with a presentation of the general implications of the results attained from the work done and possible expansions and directions for future studies as a way of continuing the discourse on the theoretical analysis and actual applications. In summary, this paper provides a useful perspective in the analysis of torsion functions solution under mixed boundary conditions which will be of interest for further development of the theory and application into practice.

In this work, we propose a path integral random walk (PIRW) solver, the first accurate stochastic method for steady-state thermal analysis with mixed boundary conditions, especially involving Fourier heat transfer Robin boundary conditions. We innovatively adopt the strictly correct calculation of the local time and the Feynman-Kac functional êc (t) to handle Neumann and Robin boundary conditions with high precision. Compared with ANSYS, experimental results show that PIRW achieves over 121× speedup and over 83× storage space reduction with a negligible error within 0.8° C at a single point. An application combining PIRW with low-accuracy ANSYS for the temperature calculation at hot-spots is provided as a more accurate and faster solution than only ANSYS used.

Transient thermal analysis captures the high accuracy and time-varying temperature distribution of chip components. However, during the early thermal design iterations of 3D ICs and chiplets, time-consuming transient thermal simulations lead to prolonged waiting times for engineers. Existing widely adopted deterministic methods or innovative neural network-based methods face significant challenges in reducing the thermal simulation time when geometric structures change iteratively. With prior knowledge of the chip’s regions of interest, localized stochastic methods may present a promising solution. In this work, we propose a novel high-precision and efficient stochastic solver for transient thermal analysis of ICs, based on the probabilistic solution of the heat equation obtained from the Feynman- Kac formula. As a field solver for the heat equation, the proposed method adopts the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spacetime</i> walk-on-sphere (WOS) scheme and enhances it with the novel boundary treatment integrated to guarantee the accuracy of addressing mixed boundary conditions, particularly Robin boundary conditions. A time-dependent fixed random walk is also combined to handle heat-source regions. Experimental results show that the proposed solver maintains high accuracy with average absolute error within 0.27 K and maximum absolute error within 1 K, achieves a speedup of 26.8-211× at a single point compared to the 3D-ICE simulator of similar accuracy, has general geometry applicability as a field solver, and exhibits excellent parallel efficiency.

This chapter provides a tutorial for the multigrid solver module in Diffpack. With the MGtools class or the NonLinMGtools class, a standard Diffpack finite element application code can be equipped with a flexible multigrid solver by adding about 10 lines of extra code. The applications covered here are the Poisson equation, more general elliptic problems with anisotropic or jumping coefficients and various boundary conditions, the equation of linear elasticity, and a nonlinear Poisson-like problem. KeywordsCoarse GridFine GridMultigrid MethodGrid LevelMenu ItemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

We investigate the asymptotic behavior of minimizers to sequences of elliptic variational problems posed on thin three-dimensional domains. These domains arise as thin neighborhoods of artibrary graphs that contain severe constrictions near the graph nodes. We characterize an appropriate limit of minimizers as a function of one variable defined on the graph that necessarily minimizes a one-dimensional variational problem. The most salient feature of these limits of minimizers is the emergence of jump discontinuities across the graph nodes. While the approach can handle quite general elliptic problems, we pay particular attention to an application to generalized Josephson junctions within the Ginzburg-Landau theory of superconductivity.