adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

an anisotropic h type mesh refinement strategy abstract an anisotropic h type mesh refinement strategy for finite element approximations is presented the strategy consists of subsecting quadrilateral elements in one of two possible directions the decision which elements should be subsected and in which direction is based on minimization of interpolation errors the method is especially suitable for approximate solutions which in some regions display almost one dimensional behavior for instance solutions with boundary layers

the discrete maximum principle for linear simplicial finite element approximations of a reaction diffusion problem abstract this paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction diffusion problem to hold it explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension as a result it can be computed how small the acute simplices should be for the discrete maximum principle to be valid numerical experiments suggest that the bound which considerably improves a similar bound in p g ciarlet p a raviart maximum principle and uniform convergence for the finite element method comput methods appl mech eng 2 1973 17 31 is in fact sharp

the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

convex hull property and maximum principle for finite element minimisers of general convex functionals the convex hull property is the natural generalization of maximum principles from scalar to vector valued functions maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model in this work we develop a convex hull property for mathbb p 1 conforming finite elements on simplicial non obtuse meshes the proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional therefore the result holds for very general nonlinear partial differential equations including e g the p laplacian and the mean curvature problem in the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems we conclude by proving a strong discrete convex hull property on strictly acute triangulations

jacobian weighted elliptic grid generation variational grid generation techniques are used to derive and analyze a weighted elliptic grid generator that controls the jacobian of the underlying transformation in a least squares sense the euler lagrange equations for the area and volume generators are weighted forms of the well known laplace generator weights are restricted to the class of p matrices to help achieve global invertibility of the map connecting the weights to the jacobian of the map results in an intuitive means of controlling grid spacing area orthogonality and grid line directions examples are given on the unit square to demonstrate point attraction local refinement directional alignment and adaption to a shock

discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures the discrete maximum principles are proved for finite element solutions of some nonlinear elliptic problems with mixed boundary conditions the effect of quadrature rules used for the construction of the stiffness matrices is taken into account

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

dissection of the path simplex in r n into n path subsimplices abstract we review properties of acute and non obtuse simplices and of ortho simplices and path simplices dissection of path simplices is considered which leads to a new result generalization of coxeter s trisection of a path tetrahedron into three path subtetrahedra to arbitrary spatial dimension n moreover following earlier results by korotov and k i ek we show that applying this procedure recursively in the proper way leads to a self similar path simplicial refinement towards a chosen vertex of the original path simplex

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

failure of the discrete maximum principle for an elliptic finite element problem there has been a long standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem this is related to knowing whether the discrete green s function is positive for triangular meshes allowing sufficiently good approximation of h 1 functions we study this question for the poisson problem in two dimensions discretized via the galerkin method with continuous piecewise linears we give examples which show that in general the answer is negative and furthermore we extend the number of cases where it is known to be positive our techniques utilize some new results about discrete green s functions that are of independent interest

a mixed implicit explicit finite difference scheme for heat transport in magnetised plasmas an explicit implicit domain decomposition method is applied to the time dependent heat conduction problem in a 2 d strongly anisotropic medium a magnetised plasma using a formulation of the spatial derivatives which avoids the pollution of perpendicular by parallel heat fluxes the time stepping at the sub domain boundaries is done using a dufort frankel scheme which leads to a time step limit given not by instabilities but by the damping rate of numerical oscillations driven by inconsistencies in the formulation of initial conditions or the temporal variations in the true physical solution these limitations can be minimized however by aligning the subdomain boundaries as much as possible with magnetic flux surfaces the time step limit depends on the ratio of the implicit grid spacing to the distance between subdomain boundaries dufort frankel lines in 2 d surfaces in 3 d

conditioning of finite element equations with arbitrary anisotropic meshes this is the published version also available here http dx doi org 10 1090 s0025 5718 2014 02822 6 first published in math comput in 2014 published by the american mathematical society

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions one of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations for problems of elliptic type one of the basic properties is the continuous maximum principle in our work we present several variants of the maximum principles and their discrete counterparts for scalar second order nonlinear elliptic problems with mixed boundary conditions the problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes sufficient conditions providing the validity of the corresponding discrete maximum principles are presented geometrically they mean that the employed meshes have to be of acute or nonobtuse type depending of the type of the problem finally some examples of real life problems where the preservation of maximum principles plays an important role are presented

monotone finite volume schemes for diffusion equations on unstructured triangular and shape regular polygonal meshes we consider a non linear finite volume fv scheme for stationary diffusion equation we prove that the scheme is monotone i e it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients the scheme is extended to regular star shaped polygonal meshes and isotropic heterogeneous coefficients

modelling of heat transport in magnetised plasmas using non aligned coordinates we present discretisation schemes for the heat diffusion equation in strongly magnetised plasmas in 2 d cartesian and 3 d cylindrical geometry the algorithms described do not use field aligned coordinates but show nevertheless a strong reduction of the pollution of perpendicular heat flux caused by parallel heat conduction compared to standard finite element or other finite difference formulations results for test examples show independence of the numerical error from g for values of g g up to 10 9 in plane and 10 1 2 in periodic cylinder geometry simulating toroidal connection the algorithms appear particularly well suited for dynamic mhd calculations where the effort of using exactly aligned coordinates becomes prohibitive

an anisotropic h adaptive finite element method for compressible navier stokes equations abstract application of the finite element method with significantly stretched elements to solve compressible navier stokes equations is presented stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements an adaptive strategy of such anisotropic h type refinements was developed it is based on minimization of the interpolation errors reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy an iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented the solver is the gmres algorithm with the domain decomposition schwarz type preconditioner the proposed techniques are used to solve a benchmark problem a hypersonic high reynolds number flow with strong shock boundary layer interaction

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

on the existence of time optimal controls with constraints of the rectangular type for heat equations this paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations an existence result of time optimal controls for such a problem is established the rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations in the finite dimensional case there is no difference between such problems with control constraints of the rectangular type and those of the ball type from the perspective of the study on the existence of optimal controls interestingly in the infinite dimensional case the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type for infinite dimensional systems the existence for time optimal controls with constraints of the ball type has already been discussed in the literature while the study of the rectangular type has not been touched upon as far as we know

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

finite element and higher order difference formulations for modelling heat transport in magnetised plasmas we present a finite element analogue to the second order finite difference scheme for the solution of the heat diffusion equation in strongly magnetised plasmas given in gunter et al s gunter et al j comp phys 209 2005 354 compared to standard finite element or finite difference formulations it strongly reduces the pollution of perpendicular heat fluxes by parallel ones even without resorting to field aligned coordinates we present both bi linear and bi quadratic versions of this scheme as well as a fourth order extension of the original difference scheme of gunter et al 2005 in the second part of the paper we address the formulation of the boundary conditions at walls with an oblique incidence of field lines and the treatment of the coordinate singularity at r 0 in cylindrical or topologically equivalent coordinates with the reduced pollution finite difference scheme all tests shown indicate that both the finite difference and the finite element versions of the scheme should substantially alleviate the requirement for field alignment of the coordinates over the realistic range of in toroidal magnetic confinement devices

anisotropic adaptive simulation of transient flows using discontinuous galerkin methods an anisotropic adaptive analysis procedure based on a discontinuous galerkin finite element discretization and local mesh modification of simplex elements is presented the procedure is applied to transient two and three dimensional problems governed by euler s equation a smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified the mesh metric field in smooth portions of the domain is controlled by a hessian matrix constructed using a variational procedure to calculate the second derivatives the transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain copyright c 2004 john wiley sons ltd

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

on the equivalence of minimal time and minimal norm controls for internally controlled heat equations we prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves we consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time as a consequence of this equivalence using the well known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

a constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems nonlinear constrained finite element approximations to anisotropic diffusion problems are considered starting with a standard linear or bilinear galerkin discretization the entries of the stiffness matrix are adjusted so as to enforce sufficient conditions of the discrete maximum principle dmp an algebraic splitting is employed to separate the contributions of negative and positive off diagonal coefficients which are associated with diffusive and antidiffusive numerical fluxes respectively in order to prevent the formation of spurious undershoots and overshoots a symmetric slope limiter is designed for the antidiffusive part the corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes the recovery of nodal gradients is performed by means of a lumped mass l 2 projection the proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix symmetry zero row and column sums a positivity preserving defect correction scheme is devised for the nonlinear algebraic system to be solved numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions

on maximum principles for monotone matrices abstract for monotone matrices and h matrices maximum principles are defined and necessary and sufficient conditions are derived for these principles to hold

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

a new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates a new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented it generates anisotropic adaptive meshes as quasi uniform ones in some metric space with the metric tensor being computed based on hierarchical a posteriori error estimates a global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution instead of solving the global error problem exactly which is costly in general we solve it iteratively using the symmetric gaus seidel method numerical results show that a few gs iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation the new method is compared with several strategies using local error estimators or recovered hessians numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients

a newton method for the computation of time optimal boundary controls of one dimensional vibrating systems abstract we consider the problem of time optimal boundary control of a one dimensional vibrating system subject to a control constraint that prescribes an upper bound for the l 2 norm of the image of the control function under a volterra operator for the solution of this problem we propose to use newton s method to compute the zero of the optimal value function of certain parametric auxiliary problems where the steering time is the parameter the formulation of the auxiliary problems which are problems of norm minimal control is based on the method of moments for a fixed parameter these problems have a simple structure we present convergence results with respect to the discretization parameters where the discretization is done by truncating the system of moment equations we prove that the optimal value function of the discretized parametric auxiliary problem is differentiable and show how the derivative can be computed so that newton s method can be used we present numerical examples for the problem of time optimal control of the rotation of an euler bernoulli beam that illustrate the fast convergence of the algorithm with respect to the time parameter

adaptive remeshing for compressible flow computations an adaptive mesh procedure for improving the quality of steady state solutions of the euler equations in two dimensions is described the procedure is implemented in conjunction with a finite element solution algorithm using linear triangular elements and an explicit time stepping scheme the mesh adaptation is accomplished by regeneration using information provided by the computed solution on the current mesh the meshes produced are characterised by the appearance of stretched elements in the vicinity of one dimensional flow features and can exhibit a large variation in element size this allows solutions of enhanced quality to be produced in a computationally efficient manner