optimal control of an eddy current problem with a dipole source abstract this paper is concerned with the analysis of a class of optimal control problems governed by a time harmonic eddy current system with a dipole source which is taken as the control variable a mathematical model is set up for the state equation where the dipole source takes the form of a dirac mass located in the interior of the conducting domain a non standard approach featuring the fundamental solution of a curl curl id operator is proposed to address the well posedness of the state problem leading to a split structure of the state field as the sum of a singular part and a regular part the aim of the control is the best approximation of desired electric and magnetic fields via a suitable l 2 quadratic tracking cost functional here special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived

splitting methods for constrained diffusion reaction systems abstract we consider lie and strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term since such systems are known to be sensitive with respect to perturbations the splitting procedure seems promising as we can treat the nonlinearity separately this has some computational advantages since we only have to solve a linear constrained system and a nonlinear ordinary differential equation however strang splitting suffers from order reduction which limits its efficiency this reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem the incorporation of an additional correction term resolves this problem without increasing the computational cost significantly numerical examples including a coupled mechanical system illustrate the proved convergence results

boundary value problems with symmetry and their approximation by finite elements symmetry and related questions of group theory especially representation theory are well known popular topics among physicists this paper describes how these classical ideas can be put to use in the realm of finite element computations by substituting a family of independent problems on a reduced domain the symmetry cell for the original problem on the whole domain two specific difficulties appear one is how to set boundary conditions on symmetry planes representation theory gives the answer the other is how to proceed with the assembly of finite elements that constitute the symmetry cell to deal with this this paper introduces the concept of index of an element as part of the whole structure which is the number of its distinct symmetrical images the rule for assembly then becomes to perform as usual but to count each element as many times as its index

optimal control of quasilinear boldsymbol h mathbf curl elliptic partial differential equations in magnetostatic field problems this paper examines the mathematical and numerical analysis for optimal control problems governed by quasilinear boldsymbol h mathbf curl elliptic partial differential equations we consider a mathematical model involving isotropic materials with magnetic permeability depending strongly on the magnetic field due to the physical and mathematical nature of the problem it is necessary to include divergence free constraints on the state and the control the divergence free control constraint is treated as an explicit variational equality constraint whereas a lagrange multiplier is included in the state equation to deal with the divergence free state constraint we investigate the sensitivity analysis of the control to state operator and establish the associated optimality conditions here the key tool for proving the kkt theory is the helmholtz decomposition an important consequence of the optimality system is a higher regularity result for the optimal control which we prove under the assumption

optimal control for electromagnetic cloaking metamaterial parameters design abstract in this paper we develop an optimal control problem to optimize the permittivity and permeability of the metamaterial which can have the invisible cloaking effect the inverse problem is calibrated as an optimization problem constrained by the time harmonic maxwell s equations proper objective functional is introduced with coefficients of the maxwell s equations as control variables we adopt the discontinuous galerkin method to solve the state equation several two dimensional cloaking benchmark problems are used to test our optimal control method numerical results demonstrate that the optimized permittivity and permeability of the metamaterial can achieve the cloaking effectively

eigenvalues of the fractional laplace operator in the interval abstract two term weyl type asymptotic law for the eigenvalues of the one dimensional fractional laplace operator 2 0 2 in the interval 1 1 is given the n th eigenvalue is equal to n 2 2 8 o 1 n simplicity of eigenvalues is proved for 1 2 l 2 and l properties of eigenfunctions are studied we also give precise numerical bounds for the first few eigenvalues

semilinear fractional elliptic equations involving measures abstract we study the existence of weak solutions to e u g u in a bounded regular domain in r n n 2 which vanish in r n where denotes the fractional laplacian with 0 1 is a radon measure and g is a nondecreasing function satisfying some extra hypotheses when g satisfies a subcritical integrability condition we prove the existence and uniqueness of weak solution for problem e for any measure in the case where is a dirac measure we characterize the asymptotic behavior of the solution when g r r k 1 r with k supercritical we show that a condition of absolute continuity of the measure with respect to some bessel capacity is a necessary and sufficient condition in order e to be solved

application of group representation theory to derive hermite interpolation polynomials on a triangle current methods used to devise sets of hermite interpolation polynomials of minimal order that ensure c n continuity across triangular element boundaries in two dimensions are not readily extensible to higher dimensions the extension of such methods is especially difficult when the number of degrees of freedom afforded by data at points is different from the number of degrees of freedom determined by the coefficients of a complete polynomial basis to a particular order this work introduces a formalism based on group representation theory that can accomplish this task in general the method is introduced through the derivation of c 1 continuous hermite polynomials that interpolate data at the three vertices of an equilateral triangular element these interpolation polynomials are reported here for the first time the polynomials derived here are compared to the standard polynomials defined in a right triangle by using the two sets of polynomials to solve the laplace equation over finite elements the methodology presented here is of use in higher dimensional elements when the complete polynomial degrees of freedom exceed the total c n degrees of freedom at the nodes

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

regularity for the fractional gelfand problem up to dimension 7 abstract we study the problem s u e u in a bounded domain r n where is a positive parameter more precisely we study the regularity of the extremal solution to this problem our main result yields the boundedness of the extremal solution in dimensions n 7 for all s 0 1 whenever is for every i 1 n convex in the x i direction and symmetric with respect to x i 0 the same holds if n 8 and s 0 28206 or if n 9 and s 0 63237 these results are new even in the unit ball b 1

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

rmcp relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics abstract a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by a relaxed mixed constraint preconditioner rmcp which is a generalization based on a parameter of the mixed constraint preconditioner mcp developed in 7 choice of optimal is driven by the spectral distribution of suitable symmetric positive definite spd matrices numerical tests performed on realistic 3d problems reveal that rmcp accelerates krylov subspace solvers by a factor up to three with respect to mcp

well posedness theory for electromagnetic obstacle problems abstract this paper develops a well posedness theory for hyperbolic maxwell obstacle problems generalizing the result by duvaut and lions 1976 5 building on the recently developed result by yousept 2020 30 we prove an existence result and study the uniqueness through a local h curl regularity analysis with respect to the constraint set more precisely every solution is shown to locally satisfy the maxwell ampere equation resp faraday equation in the region where no obstacle is applied to the electric field resp magnetic field by this property along with a structural assumption on the feasible set we are able to localize the obstacle problem to the underlying constraint regions in particular the resulting localized problem does not employ the electric test function resp magnetic test function in the area where the l 2 regularity of the rotation of the electric field resp magnetic field is not a priori guaranteed this localization strategy is the main ingredient for our uniqueness proof after establishing the well posedness we consider the case where the electric permittivity is negligibly small in the electric constraint region and investigate the corresponding eddy current obstacle problem invoking the localization strategy we derive an existence result under an l 2 boundedness assumption for the electric constraint region along with a compatibility assumption on the initial data the developed theoretical results find applications in electromagnetic shielding

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

nonlocal elliptic equations involving measures abstract in this article we study the existence of solution for the problem u f u in u 0 in r n where 0 is a parameter 0 1 and is a radon measure a weak solution is obtained by using schauder s fixed point theorem in the case where is dirac measure the symmetry of the solution is obtained by using the moving plane method