a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

block partitioned solvers for coupled poromechanics a unified framework abstract coupled poromechanical problems appear in a variety of disciplines from reservoir engineering to biomedical applications this work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations these systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations recently several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution these schemes appear distinct but a unifying feature is that they exploit the block partitioned nature of the problem to accelerate convergence this paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness we begin by focusing on two specific methods a fully implicit and a sequential implicit scheme in the first approach the system matrix is treated monolithically and a krylov iteration is used to update pressure and displacement unknowns simultaneously to accelerate convergence a preconditioning operator is introduced based on an approximate block factorization of the linear system next we analyze a sequential implicit scheme based on the fixed stress split in this method one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached we re interpret this scheme as a block preconditioned richardson iteration and we show that the preconditioning operator is identical to that used within the fully implicit approach rapid convergence in both the richardson and krylov based methods results from a particular choice for a sparse schur complement approximation this analysis leads to a unified framework for developing solution schemes based on approximate block factorizations many classic fully implicit and sequential implicit schemes are simple sub cases the analysis also highlights several new approaches that have not been previously explored for illustration we directly compare the performance and robustness of several variants on a benchmark problem

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

on the stabilization of finite volume methods with co located variables for incompressible flow finite volume methods with co located variables for incompressible flow suffer from spurious pressure oscillations unless a stabilization method is applied variations of the pressure weighed interpolation pwi method are typically used for this purpose but the pwi method does not only prevent spurious oscillations counter intuitively it also simplifies the approximation of the schur complement pressure matrix which appears in iterative solution methods such as simple

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

the dirichlet problem for the p fractional laplace equation abstract we deal with a class of equations driven by nonlocal possibly degenerate integro differential operators of differentiability order s 0 1 and summability growth p 1 whose model is the fractional p laplacian operator with measurable coefficients we review several recent results for the corresponding weak solutions supersolutions as comparison principles a priori bounds lower semicontinuity boundedness holder continuity up to the boundary and many others we then discuss the good definition of s p superharmonic functions and the nonlocal counterpart of the perron method in nonlinear potential theory together with various related results we briefly mention some basic results for the obstacle problem for nonlinear integro differential equations finally we present the connection amongst the fractional viscosity solutions the weak solutions and the aforementioned s p superharmonic functions together with other important results for this class of equations when involving general measure data and a surprising fractional version of the gehring lemma we sketch the corresponding proofs of some of the results presented here by especially underlining the development of new fractional localization techniques and other recent tools various open problems are listed throughout the paper

a parallel oseen linearized algorithm for the stationary navier stokes equations abstract based on two grid discretizations and domain decomposition a parallel oseen linearized finite element algorithm for the stationary navier stokes equations with moderate or large viscosity parameter is proposed and analyzed the key idea of the algorithm is to first solve a nonlinear problem by picard iterative method on a coarse grid and then to solve an oseen problem in parallel on a fine grid to correct the coarse grid solution by using local a priori error estimate for the finite element solution and under the uniqueness condition error bounds of the corresponding finite element solution are analyzed numerical results are also given to demonstrate the high efficiency of the algorithm

an accurate and efficient method for the incompressible navier stokes equations using the projection method as a preconditioner the projection method is a widely used fractional step algorithm for solving the incompressible navier stokes equations despite numerous improvements to the methodology however imposing physical boundary conditions with projection based fluid solvers remains difficult and obtaining high order accuracy may not be possible for some choices of boundary conditions in this work we present an unsplit linearly implicit discretization of the incompressible navier stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations since our scheme is not a fractional step algorithm it is straightforward to specify general physical boundary conditions accurately however this capability comes at the price of having to solve the time dependent incompressible stokes equations at each timestep to solve this linear system efficiently we employ a krylov subspace method preconditioned by the projection method in our implementation the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning the accuracy of the scheme is demonstrated for several problems including forced and unforced analytic test cases and lid driven cavity flows these tests consider a variety of physical boundary conditions with reynolds numbers ranging from 1 to 30000 the effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the schur complement for the time dependent incompressible stokes operator the projection method is found to be a more efficient preconditioner in most cases considered in the present work

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

principal eigenvalues of a class of nonlinear integro differential operators abstract we consider a class of nonlinear integro differential operators and prove existence of two principal half eigenvalues in bounded smooth domains with exterior dirichlet condition we then establish simplicity of the principal eigenfunctions in viscosity sense maximum principles continuity property of the principal eigenvalues with respect to domains etc we also prove an anti maximum principle and study existence result for some nonlinear problem via rabinowitz bifurcation type results

boundary regularity estimates for the nonlocal normalized p laplacian in different kind of domains abstract in this paper we establish the boundary estimates in domains with flat boundary c 1 boundary and c 1 boundary respectively for nonlocal problem of the form p s u f in and u 0 in c here p s is the nonlocal normalized p laplacian of order 2 s with s 0 1

a recurrent self evolving interval type 2 fuzzy neural network for dynamic system processing this paper proposes a recurrent self evolving interval type 2 fuzzy neural network rseit2fnn for dynamic system processing an rseit2fnn incorporates type 2 fuzzy sets in a recurrent neural fuzzy system in order to increase the noise resistance of a system the antecedent parts in each recurrent fuzzy rule in the rseit2fnn are interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang tsk type with interval weights the antecedent part of rseit2fnn forms a local internal feedback loop by feeding the rule firing strength of each rule back to itself the tsk type consequent part is a linear model of exogenous inputs the rseit2fnn initially contains no rules all rules are learned online via structure and parameter learning the structure learning uses online type 2 fuzzy clustering for the parameter learning the consequent part parameters are tuned by a rule ordered kalman filter algorithm to improve learning performance the antecedent type 2 fuzzy sets and internal feedback loop weights are learned by a gradient descent algorithm the rseit2fnn is applied to simulations of dynamic system identifications and chaotic signal prediction under both noise free and noisy conditions comparisons with type 1 recurrent fuzzy neural networks validate the performance of the rseit2fnn

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

parallel multigrid smoothing polynomial versus gauss seidel gauss seidel is often the smoother of choice within multigrid applications in the context of unstructured meshes however maintaining good parallel efficiency is difficult with multiplicative iterative methods such as gauss seidel this leads us to consider alternative smoothers we discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems two particular polynomials are considered chebyshev and a multilevel specific polynomial the advantages of polynomial smoothing over traditional smoothers such as gauss seidel are illustrated on several applications poisson s equation thin body elasticity and eddy current approximations to maxwell s equations while parallelizing the gauss seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates we show that although parallel computers are the main motivation polynomial smoothers are often surprisingly competitive with gauss seidel smoothers on serial machines

min max formulas for nonlocal elliptic operators on euclidean space abstract an operator satisfies the global comparison property if anytime a function touches another from above at some point then the operator preserves the ordering at the point of contact this is characteristic of degenerate elliptic operators including nonlocal and nonlinear ones in previous work the authors considered such operators in riemannian manifolds and proved they can be represented by a min max formula in terms of levy operators in this note we revisit this theory in the context of euclidean space with the intricacies of the general riemannian setting gone the ideas behind the original proof of the min max representation become clearer moreover we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

new preconditioning techniques for the steady and unsteady buoyancy driven flow problems abstract in this paper we study the performances of generalized minimal residual method gmres preconditioned with geometric multigrid gmg applied to steady and unsteady buoyancy driven flow problems discretized with the finite element method for the unsteady case the second order crank nicolson method is used for the temporal discretization at each geometric multigrid level we use richardson iterative solvers preconditioned with different combinations of physics based and domain decomposition preconditioners three different preconditioners are considered incomplete lu decomposition ilu overlapping vanka type domain decomposition for additive schwarz method asm and field split fs physics based decomposition we also analyze the effect on the smoother of how the variables are ordered and in particular whether the leading variable is the velocity or the temperature resulting in six classes of preconditioners ilu vt ilu tv asm vt asm tv fs vt and fs tv the eigenvalue analysis for the six preconditioners is conducted to study the rate of gmres convergence under several prandtl numbers the numerical performances of nested combinations of the above preconditioners are compared numerical results show that the pair of fs vt and fs tv preconditioners works better than the other two pairs and that the fs tv preconditioner always performs the best in terms of the computational time for all the steady and unsteady cases

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

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

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

a new algebraic multigrid approach for stokes problems standard discretizations of stokes problems lead to linear systems of equations in saddle point form making difficult the application of algebraic multigrid methods in this paper a new approach is proposed it consists in first transforming the system by pre and post multiplication with simple algebraic sparse block triangular matrices this is a form of pre conditioning in the literal sense designed to make sure that the transformed matrix is well adapted to multigrid in particular after transformation all the diagonal blocks are symmetric and positive definite and correspond to or resemble a discrete laplace operator then to each of these diagonal blocks is associated a prolongation that works well for it using any relevant algebraic or geometric multigrid method next a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a galerkin coarse grid matrix for this approach combined with damped jacobi smoothing a uniform two grid convergence bound is derived for the global system under the assumption that the two grid schemes for the different diagonal blocks are themselves uniformly convergent this result is illustrated by a few examples showing further that time dependent problems and variable viscosity can be handled in a natural way without requiring parameter adjustment a numerical comparison also shows that the new approach can be more effective than state of the art block preconditioning techniques

traveling wave solutions for bistable fractional allen cahn equations with a pyramidal front abstract using the method of sub super solution we construct a solution of s u c u z f u 0 on r 3 of pyramidal shape here s is the fractional laplacian of sub critical order 1 2 s 1 and f is a bistable nonlinearity hence the existence of a traveling wave solution for the parabolic fractional allen cahn equation with pyramidal front is asserted the maximum of planar traveling wave solutions in various directions gives a sub solution a super solution is roughly defined as the one dimensional profile composed with the signed distance to a rescaled mollified pyramid in the main estimate we use an expansion of the fractional laplacian in the fermi coordinates

chaotic orbits for systems of nonlocal equations we consider a system of nonlocal equations driven by a perturbed periodic potential we construct multibump solutions that connect one integer point to another one in a prescribed way in particular heteroclinic homoclinic and chaotic trajectories are constructed this is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

least squares preconditioners for stabilized discretizations of the navier stokes equations this paper introduces two stabilization schemes for the least squares commutator lsc preconditioner developed by elman howle shadid shuttleworth and tuminaro siam j sci comput 27 2006 pp 1651 1668 for the incompressible navier stokes equations this preconditioning methodology is one of several choices that are effective for navier stokes equations and it has the advantage of being defined from strictly algebraic considerations it has previously been limited in its applicability to div stable discretizations of the navier stokes equations this paper shows how to extend the same methodology to stabilized low order mixed finite element approximation methods

factorization methods for the numerical approximation of navier stokes equations we investigate a general approach for the numerical approximation of incompressible navier stokes equations based on splitting the original problem into successive subproblems which are cheaper to solve the splitting is obtained through an algebraic approximate factorization of the matrix arising from space and time discretization of the original equations several schemes based on approximate factorization are investigated for some of these methods a formal analogy with well known time advancing schemes such as the projection chorin temam s can be pointed out features and limits of this analogy that was earlier introduced in b perot j comp phys 108 1993 51 8 are addressed other new methods can also be formulated starting from this approach in particular we introduce the so called yosida method which can be investigated in the framework of quasi compressibility schemes numerical results illustrating the different performances of the different methods addressed are presented for a couple of test cases

block recursive lu preconditioners for the thermally coupled incompressible inductionless mhd problem the thermally coupled incompressible inductionless magnetohydrodynamics mhd problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling this system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest therefore fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand however solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners in this work a new family of recursive block lu preconditioners is designed and tested for solving the thermally coupled inductionless mhd equations these preconditioners are obtained after splitting the fully coupled matrix into one physics problems for every variable velocity pressure current density electric potential and temperature that can be optimally solved e g using preconditioned domain decomposition algorithms the main idea is to arrange the original matrix into an arbitrary 2x2 block matrix and consider an lu preconditioner obtained by approximating the corresponding schur complement for every one of the diagonal blocks in the lu preconditioner if it involves more than one type of unknowns we proceed the same way in a recursive fashion this approach is stated in an abstract way and can be straightforwardly applied to other multiphysics problems further we precisely explain a flexible and general software design for the code implementation of this type of preconditioners

a parallel two level finite element variational multiscale method for the navier stokes equations abstract a combination method of two grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible navier stokes equations is proposed and analyzed the method consists of a global small scale nonlinear navier stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains where the numerical form of the navier stokes equations on the coarse grid is stabilized by a stabilization term based on two local gauss integrations at element level and defined by the difference between a consistent and an under integrated matrix involving the gradient of velocity by the technical tool of local a priori estimate for the finite element solution error bounds of the discrete solution are estimated algorithmic parameter scalings are derived numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

further time regularity for nonlocal fully nonlinear parabolic equations we establish holder estimates for the time derivative of solutions of nonlocal parabolic equations under mild assumptions for the boundary data as a consequence we are able to extend the evans krylov estimate for rough kernels to parabolic equations 2016 wiley periodicals inc

a dirichlet problem for nonlocal degenerate elliptic operators with internal nonlinearity abstract we study a dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities with very mild assumptions on the boundary datum we prove existence and uniqueness of the solution in the viscosity sense if we further assume uniform ellipticity then the solution is shown to be classical and even smooth if both the operator and the boundary datum are smooth

large solutions for a class of semilinear integro differential equations with censored jumps we study existence of large solutions that is solutions that verify u x 1 as x for equations like i u x u x p 0 x2 where is a bounded smooth domain in r n p 1 andi is a nonlocal operator of the form i u x p v z jzj x u x z u x jzj n dz where 2 0 2 and r is a function whose main particularity is that 0 x dist x we also obtain uniqueness of the solution in a class of large solutions whose blow up rate depends on p and the rate at which shrinks near the boundary

a massively parallel fractional step solver for incompressible flows this paper presents a parallel implementation of fractional solvers for the incompressible navier stokes equations using an algebraic approach under this framework predictor corrector and incremental projection schemes are seen as sub classes of the same class making apparent its differences and similarities an additional advantage of this approach is to set a common basis for a parallelization strategy which can be extended to other split techniques or to compressible flows the predictor corrector scheme consists in solving the momentum equation and a modified continuity equation namely a simple iteration for the pressure schur complement consecutively in order to converge to the monolithic solution thus avoiding fractional errors on the other hand the incremental projection scheme solves only one iteration of the predictor corrector per time step and adds a correction equation to fulfill the mass conservation as shown in the paper these two schemes are very well suited for massively parallel implementation in fact when compared with monolithic schemes simpler solvers and preconditioners can be used to solve the non symmetric momentum equations gmres bi cgstab and to solve the symmetric continuity equation cg deflated cg this gives good speedup properties of the algorithm the implementation of the mesh partitioning technique is presented as well as the parallel performances and speedups for thousands of processors

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

holder estimates for non local parabolic equations with critical drift abstract in this paper we extend previous results on the regularity of solutions of integro differential parabolic equations the kernels are non necessarily symmetric which could be interpreted as a non local drift with the same order as the diffusion we provide a growth lemma and a harnack inequality which can be used to prove higher regularity estimates

on the spectrum of two different fractional operators in this paper we deal with two non local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type more precisely for a fixed s 0 1 we consider the integral definition of the fractional laplacian given by where c n s is a positive normalizing constant and another fractional operator obtained via a spectral definition that is where e i i are the eigenfunctions and the eigenvalues of the laplace operator in with homogeneous dirichlet boundary data while a i represents the projection of u on the direction e i the aim of this paper is to compare these two operators with particular reference to their spectrum in order to emphasize their differences

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

a block preconditioner for an exact penalty formulation for stationary mhd the magnetohydrodynamics mhd equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas this system of non self adjoint nonlinear pdes couples the navier stokes equations for fluids and maxwell s equations for electromagnetics there has been recent interest in fully coupled solvers for the mhd system because they allow for fast steady state solutions that do not require pseudo time stepping when the fully coupled system is discretized the strong coupling can make the resulting algebraic systems difficult to solve requiring effective preconditioning of iterative methods for efficiency in this work we consider a finite element discretization of an exact penalty formulation for the stationary mhd equations posed in two dimensional domains this formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a lagrange multiplier we consider extending block preconditioning techni

interior regularity for nonlocal fully nonlinear equations with dini continuous terms abstract this paper is concerned with interior regularity of viscosity solutions of non translation invariant nonlocal fully nonlinear equations with dini continuous terms we obtain c regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive evans krylov theorem

facsi a block parallel preconditioner for fluid structure interaction in hemodynamics modeling fluid structure interaction fsi in the vascular system is mandatory to reliably compute mechanical indicators in vessels undergoing large deformations in order to cope with the computational complexity of the coupled 3d fsi problem after discretizations in space and time a parallel solution is often mandatory in this paper we propose a new block parallel preconditioner for the coupled linearized fsi system obtained after space and time discretization we name it facsi to indicate that it exploits the factorized form of the linearized fsi matrix the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations and the use of a simple preconditioner for saddle point problems facsi is built upon a block gauss seidel factorization of the fsi jacobian matrix and it uses ad hoc preconditioners for each physical component of the coupled problem namely the fluid the structure and the geometry in the fluid subproblem after operating static condensation of the interface fluid variables we use a simple preconditioner on the reduced fluid matrix moreover to efficiently deal with a large number of processes facsi exploits efficient single field preconditioners e g based on domain decomposition or the multigrid method we measure the parallel performances of facsi on a benchmark cylindrical geometry and on a problem of physiological interest namely the blood flow through a patient specific femoropopliteal bypass we analyze the dependence of the number of linear solver iterations on the cores count scalability of the preconditioner and on the mesh size optimality c 2016 elsevier inc all rights reserved

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

an overlapping domain decomposition preconditioning method for monolithic solution of shear bands metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material s load bearing capability a ductile failure phenomenon known as shear banding shear bands characterized as material instabilities are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise this coupled thermo mechanical localization problem can be formulated as a nonlinear system with two balance equations momentum and energy and two constitutive laws for elasticity and plasticity in our formulation mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time where at every time step a newton type method is used to solve the system monolithically to that end a block jacobian matrix is formed analytically using g teaux derivatives the resulting jacobian is sparse nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation consequently posing a significant challenge to iterative solvers to address this issue an overlapping domain decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms is proposed the key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately with selective updates when necessary the preconditioner is formed on the basis of an additive schwartz method that is applied to a schur complement partition of the system the proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off the shelf solvers we study the effect of hh and kk refinement that are obtained from isogeometric discretizations of the system the overlapping strategy and its parallel performance excellent performance is demonstrated in serial and parallel on all benchmark examples

obstacle problems for integro differential operators regularity of solutions and free boundaries we study the obstacle problem for integro differential operators of order 2s with s in 0 1 our main result establish that the free boundary is c 1 gamma and u in c 1 s near all regular points namely we prove the following dichotomy at all free boundary points x 0 in partial u varphi i either u x varphi x c d 1 s x o x x 0 1 s alpha for some c 0 ii or u x varphi x o x x 0 1 s alpha where d is the distance to the contact set u varphi moreover we show that the set of free boundary points x 0 satisfying i is open and that the free boundary is c 1 gamma and u in c 1 s near those points these results were only known for the fractional laplacian 2 and are completely new for more general integro differential operators the methods we develop here are purely nonlocal and do not rely on any monotonicity type formula for the operator thanks to this our techniques can be applied in the much more general context of fully nonlinear integro differential operators we establish similar regularity results for obstacle problems with convex operators

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

sleipnnir a multiscale particle level set method for newtonian and non newtonian interface flows abstract we present in this paper a multiscale micro macro particle level set method for newtonian and non newtonian interface flows the technique termed sleipnnir semi lagrangian ensemble implementation of particle level set for newtonian and non newtonian interfacial rheology uses the finite element method in a semi lagrangian framework for the discretization of the governing equations capturing the free surface by means of a particle level set strategy where all involved magnitudes are computed sharply surface tension effects are considered via the laplace beltrami operator using a fully explicit or semi implicit technique with a second order accurate reinitialization procedure to ensure the signed distance property non newtonian fluids are tackled by means of a stochastic variance reduced kinetic modeling approach in which the polymer stress tensor is retrieved making use of compactly supported radial basis functions for randomly scattered data interpolation we carry out numerical tests to highlight the versatility robustness and accuracy of the proposed scheme through a series of bubble dynamics experiments in a 2d setup for large density and viscosity ratios and featuring strong purely elastic effects relevant to engineering applications

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

unified computational framework for the efficient solution of n field coupled problems with monolithic schemes abstract in this paper we propose and evaluate the performance of a unified computational framework for preconditioning systems of linear equations resulting from the solution of coupled problems with monolithic schemes the framework is composed by promising application specific preconditioners presented previously in the literature with the common feature that they are able to be implemented for a generic coupled problem involving an arbitrary number of fields and to be used to solve a variety of applications the first selected preconditioner is based on a generic block gauss seidel iteration for uncoupling the fields and standard algebraic multigrid amg methods for solving the resulting uncoupled problems the second preconditioner is based on the semi implicit method for pressure linked equations simple which is extended here to deal with an arbitrary number of fields and also results in uncoupled problems that can be solved with standard amg finally a more sophisticated preconditioner is considered which enforces the coupling at all amg levels in contrast to the other two techniques which resolve the coupling only at the finest level our purpose is to show that these methods perform satisfactory in quite different scenarios apart from their original applications to this end we consider three very different coupled problems thermo structure interaction fluid structure interaction and a complex model of the human lung numerical results show that these general purpose methods are efficient and scalable in this range of applications

a fractional step method for steady state flow abstract a fractional step method for solving the steady state incompressible navier stokes equations is presented the proposed iterative method uses an estimate for the optimal under relaxation coefficient at each iteration this estimate uses prior information that is already available in the iterative method so the calculation has negligible cost numerical tests show that the convergence rate of the fractional step method using this optimal relaxation parameter is nearly independent of the mesh size resulting in significant cost savings for problems with fine meshes the ideas demonstrated in this paper are general and can be used to improve many existing iterative methods for steady flows as well as methods for the unsteady navier stokes equations and methods for other saddle point problems

regularity results for nonlocal equations by approximation we obtain c1 regularity estimates for nonlocal elliptic equations that are not necessarily translation invariant using compactness and perturbative methods and our previous regularity results for the translation invariant case

multi space reduced basis preconditioners for parametrized stokes equations abstract we introduce a two level preconditioner for the efficient solution of large scale saddle point linear systems arising from the finite element fe discretization of parametrized stokes equations this preconditioner extends the multi space reduced basis msrb preconditioning method proposed in dal santo et al 2018 it combines an approximated block fine grid preconditioner with a reduced basis rb solver which plays the role of coarse component a sequence of rb spaces constructed either with an enriched velocity formulation or a petrov galerkin projection is built each rb coarse component is defined to perform a single iteration of the iterative method at hand the flexible gmres fgmres algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time numerical test cases for stokes flows in three dimensional parameter dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings