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

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

an iterative method for the boundary element solution of the exterior acoustic problem abstract in this paper we discuss the burton and miller s boundary integral formulation of the exterior neumann problem for the helmholtz equation which is uniquely solvable for all frequencies as well as a similar formulation of brakhage and werner lies and also panich for the exterior dirichlet problem we present an iterative procedure for the efficient solution of the full linear systems arising from the discretisation of such boundary integral equations numerical results will also be presented for the neumann problem for a class of axisymmetric ellipsoids

on multilevel iterative methods for integral equations of the second kind and related problems we describe a unifying framework for multigrid methods and projection iterative methods for integral equations of the second kind and for the iterative aggregation method for solving input output relations the methods are formulated as iterations combined with a defect correction in a subspace convergence proofs use contraction arguments and thus involve the nonlinear case automatically some new results are presented

multiple grid methods for the solution of fredholm integral equations of the second kind in this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of fredholm integral equations of the second kind various multiple grid schemes both with nystrom and with direct interpolation are considered for these iterative methods the rates of convergence are derived using the collectively compact operator theory by anselone and atkinson estimates for the asymptotic computational complexity are given which show that the multiple grid schemes result in e n2 arithmetic operations

two way coupling in reservoir geomechanical models vertex centered galerkin geomechanical model cell centered and vertex centered finite volume reservoir models procedures to couple reservoir and geomechanical models are reviewed the focus is on immiscible compressible non compositional reservoir geomechanical models such models require the solution to coupled stress pressure saturation and temperature equations although the couplings between saturation and temperature with stress and fluid pressure are weak and can be adequately captured thru staggered fixed point iterations the couplings between stress and pressure equations are strong and require special procedures for accurate integration as shown and discussed in detail in prevost 2013 two way coupling i e simultaneous integration of pressure and stress equations is required if poromechanical effects are to be captured accurately in our previous work a galerkin implementation of both pressure and stress equations was used with equal order interpolant for both pressure and solid displacements

fast solution methods for fredholm integral equations of the second kind the main purpose of this paper is to describe a fast solution method for one dimensional fredholm integral equations of the second kind with a smooth kernel and a non smooth right hand side function let the integral equation be defined on the interval 1 1 we discretize by a nystrom method with nodes cos j n j 0 n this yields a linear system of algebraic equations with an n 1 n 1 matrixa generallyn has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation we show by fourier analysis thata can be approximated well by mathop a limits sim a low rank modification of the identity matrix replacinga by mathop a limits sim in the linear system of algebraic equations yields a new linear system of equations whose elements and whose solution tilde x can be computed ino n logn arithmetic operations if the kernel has two more derivatives than the right hand side function then tilde x is shown to converge optimally to the solution of the integral equation asn increases we also consider iterative solution of the linear system of algebraic equations the iterative schemes use botha anda they yield the solution ino n 2 arithmetic operations under mild restrictions on the kernel and the right hand side function finally we discuss discretization by the chebyshev galerkin method the techniques developed for the nystrom method carry over to this discretization method and we develop solution schemes that are faster than those previously presented in the literature the schemes presented carry over in a straightforward manner to fredholm integral equations of the second kind defined on a hypercube

a fast boundary element method for the two dimensional helmholtz equation abstract in this paper we present a fast method for solving boundary integral equations arising from the exterior dirichlet problem for the two dimensional helmholtz equation this method combines a quadrature method for discretizing the boundary integral equations with a preconditioned iterative method for solving the resulting dense nonsymmetric linear systems using this method a polynomial rate of convergence can be obtained by performing a finite number of iterations which yields high computational efficiency various numerical examples are presented

stabilized mixed continuous enriched galerkin formulations for locally mass conservative poromechanics abstract local element wise mass conservation is often highly desired for numerical solutions to coupled poromechanical problems as an efficient numerical method featuring this property mixed continuous galerkin cg enriched galerkin eg finite elements have recently been proposed in which piecewise constant functions are enriched to the pore pressure interpolation functions of the conventional mixed cg cg elements while this enrichment of the pressure space provides local mass conservation it unavoidably alters the stability condition for mixed finite elements because no stabilization method has been available for the new stability condition high order displacement interpolation has been required for mixed cg eg elements if undrained condition is expected to circumvent this requirement here we develop stabilized formulations for the mixed cg eg elements that permit equal order interpolation functions even in the undrained limit we begin by identifying the inf sup condition for mixed cg eg elements by phrasing an enriched poromechanical problem as a twofold saddle point problem we then derive two types of stabilized formulations one based on the polynomial pressure projection ppp method and the other based on the fluid pressure laplacian fpl method a key finding of this work is that both methods lead to stabilization terms that should be augmented only to the cg part of the pore pressure field not to the enrichment part the two stabilized formulations are verified and investigated through numerical examples that involve various conditions spanning 1d to 3d isotropy to anisotropy and homogeneous to heterogeneous domains the methodology presented in this work may also help stabilize other types of mixed finite elements in which the constraint field is enriched by additional functions

multilevel algorithms for ill posed problems in this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations convergence estimates are established for multilevel algorithms applied to tikhonov type regularization methods our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter the algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations

spectrally accurate nystrom solver error bounds for 1 d fredholm integral equations of the second kind abstract we present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one dimensional fredholm integral equations of the second kind through asymptotic estimates of near supremal operator norms readily implementable formulae for the error bounds are computed explicitly using only the numerical solution of nystrom based methods on distributions of nodes at the roots or extrema of diverse orthogonal polynomials despite the predicted bounds demanding no a priori information about the exact solution they are validated to be spectrally accurate upon comparison with the explicit computational error accruing from the numerical solution of a variety of test problems some chosen to be challenging to approximation methods with known solutions potential limitations of the theory are discussed but these are shown not to arise in the numerical computations

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

stability and convergence of sequential methods for coupled flow and geomechanics fixed stress and fixed strain splits abstract we analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics in which the flow problem is solved first we employ the von neumann and energy methods for linear and nonlinear problems respectively we consider two sequential methods with the generalized midpoint rule for tn where is the parameter of time discretization namely the fixed strain and fixed stress splits the von neumann method indicates that the fixed strain split is only conditionally stable and that its stability limit is a coupling strength less than unity if 0 5 on the other hand the fixed stress split is unconditionally stable when 0 5 the amplification factors of the fixed stress split are different from those of the undrained split and are identical to the fully coupled method unconditional stability of the fixed stress split is also obtained from the energy method for poroelastoplasticity we show that the fixed stress split is contractive and b stable when 0 5 we also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward euler method in time from the estimates the fixed strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable the fixed stress split however is convergent for a fixed number of iterations showing better accuracy than the undrained split even when we cannot obtain the exact local bulk modulus or exact rock compressibility at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials the fixed stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus such as the dimension based estimation by which the employed local bulk modulus is less stiff than the exact local bulk modulus we provide numerical examples supporting all the estimates of stability and convergence for the fixed strain and fixed stress splits

application of multigrid methods for integral equations to two problems from fluid dynamics abstract multigrid methods are applied in order to solve efficiently the nonsparse systems of equations that occur in the numerical solution of the following problems from fluid dynamics 1 calculation of potential flow around bodies and 2 calculation of oscillating disk flow problem 1 is reformulated as a boundary integral equation of the second kind that is approximated by a first order panel method resulting in a full system of equations this method is in widespread use for aerodynamic computations the second problem is described by the navier stokes and continuity equations by means of the von karman similarity transformations these equations are reduced to a nonlinear system of parabolic equations which are approximated by implicit finite difference techniques from the periodic conditions in time one obtains a nonsparse system of equations for these two problems from fluid dynamics the fast convergence of multigrid methods for integral equations is established by numerical experiments

a multilevel formulation of the method of moments em scattering the modeling of electromagnetic scattering and radiation problems using the method of moments mom is limited to resonant frequencies because of the extensive computational requirements of solving large matrix equations in this study a multilevel formulation of mom which allows substantial computational savings and thus extends the application of mom to higher frequencies is presented using a hierarchy of discretization levels the multilevel technique extracts different modal components of the solution by focusing on a specific portion of the spectrum of the solution at a given level the fundamental features of this process for the mom solution of the electric field integral equation efie are developed and implemented this multilevel mom allows the rapid evaluation of the current distributions on a variety of 2 d scatterers with thousands of unknowns in fewer than ten cycles and in fractions of the normal cpu times the method is stable fast suitable for multiple excitations and adaptable as a solve module for almost any mom code

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

discretization of the timoshenko beam problem by thep and theh p versions of the finite element method in this paper the discretization of the timoshenko beam problem by thep and theh p versions of the finite element method is considered optimal error estimates are established the locking phenomenon disappears as the thickness of the beam decreases

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

multigrid methods for boundary integral equations multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind these methods originally formulated for partial differential equations of elliptic type combine relaxation schemes and coarse grid corrections the choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid this is illustrated for the calculation of potential flow around an aerofoil

conjugate gradient methods for the solution of boundary integral equations on a piecewise smooth boundary abstract the conjugate gradient methods cgms have been successfully applied to solve the complex matrix equations arising from discretization of boundary integral equations if the underlying integral operator is compact its eigenvalue clustering property ensures the fast convergence of these methods such an integral operator is usually compact if the integral boundary is globally smooth in this paper however we consider the numerical solution of the boundary integral equation with a non compact operator where the non compactness is due to the non smoothness of a pieceiwe smooth boundary two particular algorithms are presented and tested for a model problem we show that such non compact integral equations can be solved efficiently by the preconditioned conjugate gradient method and that the algorithm using the normal equation appears to be particularly efficient