a bie method for a nonlinear bvp abstract a nonlinear boundary value problem bvp governed by laplace s equation with a nonlinear boundary condition is considered this paper is concerned with the numerical solvability of a nonlinear boundary integral equation bie obtained by reformulating the nonlinear bvp we give a simple alternative to the standard collocation method for the nonlinear bie which requires less computational effort the method is a bie analogue of the product approximation method used for solving some nonlinear problems by the finite element method we obtain superconvergence of approximate solutions introduced in this paper

a numerical upscaling method for an elliptic equation with heterogeneous tensorial coefficients the aim of this paper is to develop a new numerical method to determine the effective permeability on a coarse scale level of problems with strongly heterogeneous tensorial coefficients defined on a fine scale this method stems from a primal dual mixed formulation and the upscaled permeability is obtained by imposing the continuity of the numerical flux across the lines of discontinuities moreover a finite volume scheme is built to numerically solve elliptic equations with this kind of coefficients finally the method is illustrated by several numerical experiments copyright 2002 john wiley sons ltd

a domain decomposition method for domains with holes using a complementary decomposition in this paper a new non conforming domain decomposition method based on fictitious domains is considered for linear elasticity problems on domains with holes existence and uniqueness of the solution of this alternative approach are showed in the continuous and discrete case although the established a priori estimate for the discretization error is not optimal the scheme is of interest in case of moving geometries and in shape optimization to solve the arising coupled linear system we use a block sor method the convergence rate does not depend on the mesh size numerical results illustrate the flexibility of the discretization scheme

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

estimate of morse index of cooperative elliptic systems and its application to spatial vector solitons local instability index of unstable solutions to single partial differential equations pdes by a local minimax method lmmm was established in zhou 2005 it is known that the local min orthogonal method lmom which was first proposed in zhou 2004 and then further developed in chen et al 2008 can find more general unstable solutions to both single pdes and cooperative elliptic systems this paper is to carry out instability analysis of unstable solutions by lmom to which an infinite dimensional functional space can be decomposed as a direct sum of a finite dimensional subspace and its orthogonal complement a morse index approach is developed to show that with lmom instability behavior of a solution in such infinite dimensional complement subspace can be totally determined usual instability analysis in an entire space is then reduced to analysis in its finite dimensional subspace for which a corresponding matrix decomposition is proposed to analyze a solution s instability behavior estimates of morse index are also established finally numerical examples of both 2 and 3 component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons whose local instabilities are numerically confirmed by the new estimates certain important properties of the examples are also verified or presented

convergence behavior of generalized parameterized uzawa method for singular saddle point problems in this paper we will seek the least squares solution for singular saddle point problems the parameterized uzawa pu method is further studied and a generalized pu gpu proper splitting is proposed the convergence behavior of the corresponding gpu iteration is studied it is proved that the gpu iteration method can converge to the best least squares solutions of the singular saddle point problems in addition we prove that the gpu preconditioned gmres for singular saddle point problems will also determine the least squares solution at breakdown the eigenvalue distributions of the gpu preconditioned matrix are derived numerical experiments are presented which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the gpu iteration has better convergence behavior than the pu iteration both as a solver and a preconditioner of gmres

reduced basis approximation of parametrized optimal flow control problems for the stokes equations this paper extends the reduced basis method for the solution of parametrized optimal control problems presented in negri et al 2013 to the case of noncoercive elliptic equations such as the stokes equations we discuss both the theoretical properties with particular emphasis on the stability of the resulting double nested saddle point problems and on aggregated error estimates and the computational aspects of the method then we apply it to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three dimensional flow through boundary control demonstrating the effectivity of the methodology

a new augmented singular transform and its partial newton correction method for finding more solutions to nonvariational quasilinear elliptic pdes abstract in this paper in order to find more solutions to a nonvariational quasilinear pde a new augmented singular transform ast is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution thus a solution found by the algorithm must be new mathematical justifications of ast formulation are established a partial newton correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in ast the new method is applied to numerically investigate bifurcation symmetry breaking phenomena to a non variational quasilinear elliptic equation through finding multiple solutions such phenomena are numerically captured and visualized for the first time and still open for theoretical verification since the formulation is general and simple it opens a door to solve other multiple solution problems

mixed hp finite element methods for stokes and non newtonian flow abstract we analyze the stability of hp finite elements for viscous incompressible flow for the classical velocity pressure formulation we give new estimates for the discrete inf sup constants on geometric meshes which are explicit in the polynomial degree k of the elements in particular we obtain new bounds for p elements on triangles for the three field stokes problem describing linearized non newtonian flow we estimate discrete inf sup constants explicit in both h and k for various subspace choices continuous and discontinuous for the extra stress we also give a stability analysis of the hp version of an elastic viscous split stress evss method and present elements that are stable and optimal in h and k finally we present numerical results that show the exponential convergence of the hp version for stokes flow over unsmooth domains

on pss based constraint preconditioners for singular nonsymmetric saddle point problems for singular nonsymmetric saddle point problems a class of singular constraint preconditioners and the corresponding preconditioning iteration methods are proposed based on the positive definite and skew symmetric splitting pss of 1 1 block of the coefficient matrix which can be categorized into a recently proposed generalized constraint preconditioner gcp for singular nonsymmetric saddle point problems the convergence analysis of the preconditioned iteration method is presented and the convergence conditions are derived some numerical experiments are implemented to demonstrate the feasibility and effectiveness of the pss based constraint preconditioning both as a preconditioned iteration method and a preconditioner of the krylov subspace method

mixed approximations for boundary elements we describe a mixed boundary element discretization of the steklov poincare operator related to a three dimensional boundary value problem with mixed boundary conditions of dirichlet and neumann type the galerkin discretization of the steklov poincare operator is replaced by a mixed approximation based on a boundary integral equation including single and double layer potentials only where appropriate stability conditions have to be assumed we show that the required stability conditions hold when either isoparametric trial functions are used or when the approximation of the steklov poincare operator is done on a refined boundary element mesh a numerical example for a mixed boundary value problem for the laplace equation confirms the theoretical results

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

finite element approximations of viscous flows with varying density this paper deals with the equations of a stationary viscous flow when the density is a given function of the space variable two formulations are studied according to whether the unknowns are the velocity and the pressure or the momentum and the pressure in both cases when the variation of the density is not too strong the well posedness of the problem and of its approximation by several standard finite elements is proved and estimates for the error between the exact and discrete solutions are given some numerical tests which are in complete agreement with these results are presented

on conforming mixed finite element methods for incompressible viscous flow problems abstract the asymptotic rates of convergence for approximate solutions of linearizations of the stationary navier stokes equations are computationally determined for some specific choices of conforming finite element spaces these rates are computed for norms of physical interest and are compared to available theoretical estimates it is shown that equivalent rates of convergence are achieved by algorithms which differ greatly in their computer storage and time requirements the solution of the discrete system of equations resulting from the finite element discretization is also discussed

a numerical investigation of sign changing solutions to superlinear elliptic equations on symmetric domains in this paper we investigate numerically sign changing solutions of superlinear elliptic equations on symmetric domains based upon the symmetric criticality principle of palais the existence of sign changing solutions which re2ect the symmetry

two classes of ljusternik schnirelman minimax algorithms and an application for finding multiple negative energy solutions of a class of p laplacian equations abstract in this paper two classes of ls minimax algorithms are presented they are applied to numerically find multiple negative energy solutions of the p laplacian equation p u u r 1 u u q 1 u x r l u 0 x where is an open bounded domain 0 r p 1 q p 1 0 and p is the sobolev exponent and mathematical justification and global convergence result for them are established by combining ls minimax algorithm with the finite element method it is verified that as element size goes to zero numerical solutions of p laplacian equation captured by ls minimax algorithm converge to solutions of p laplacian equation two ls minimax algorithms developed in 1 are two special algorithms in these two classes of algorithms

quasi optimal approximation of surface based lagrange multipliers in finite element methods we show quasi optimal a priori convergence results in the l 2 and h 1 2 norm for the approximation of surface based lagrange multipliers such as those employed in the mortar finite element method we improve on the estimates obtained in the standard saddle point theory where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estimates for the dual variable are reached for the lowest order case i e k 1 an additional factor of sqrt h ln h and for higher order cases i e k 1 an additional factor of sqrt h in the a priori bound for the dual variable can be recovered the proof is based on the use of new estimates for the primal variable in strips of width o h near these surfaces

a local min orthogonal method for finding multiple saddle points abstract the purpose of this paper is twofold the first is to remove a possible ill posedness related to a local minimax method developed in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 and the second is to provide a local characterization for nonminimax type saddle points to do so a local l selection is defined and a necessary and sufficient condition for a saddle point is established which leads to a min orthogonal method those results exceed the scope of a minimax principle the most popular approach in critical point theory an example is given to illustrate the new theory with this local characterization the local minimax method in siam j sci comput 23 2001 840 865 siam j sci comput 24 2002 840 865 is generalized to a local min orthogonal method for finding multiple saddle points in a subsequent paper this approach is applied to define a modified pseudo gradient flow of a functional for finding multiple saddle points in banach spaces

birational automorphisms of fano hypersurfaces v v y v 0 between smooth three dimensional quartics v v4 v 0 v 0 4 in p is a projective isomorphism this is the original wording of their theorem in fact the arguments of im give a much stronger result namely they actually prove that smooth quartics are birationally superrigid the concept of birational rigidity is not visual being rather technical it makes little impression it is immediate implications of the property of being birationally super rigid that are really impressive for a projective variety x smooth in codimension 1 and a linear system jdj free from xed components we de ne the threshold of canonical adjunction or brie y just threshold of the pair x d setting

a connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

stabilized finite elements for a reaction dispersion saddle point problem with nonconstant coefficients we consider a system of two reaction dispersion equations with nonconstant parameters both equations are coupled through the boundary conditions we propose a mixed variational formulation that leads to a nonsymmetric saddle point problem we prove its well posedness then we develop a stabilized mixed finite element discretization of this problem and establish optimal a priori error estimates

coupling of darcy forchheimer and compressible navier stokes equations with heat transfer this paper is devoted to the coupling of a 2d reservoir model with a 1 5d vertical wellbore model both written in axisymmetric form the physical problems are respectively described by the darcy forchheimer and the compressible navier stokes equations together with an exhaustive energy equation each model was previously studied and its finite element discretization was validated the two weak problems are bound together by means of transmission conditions at the perforations yielding a nonstandard mixed formulation a technical analysis is then carried out and the well posedness of the time discretized coupled problem in both the continuous and the discrete cases is established numerical tests including physical cases are presented validating the coupled code

existence uniqueness and approximation for generalized saddle point problems extensions of the well known results of brezzi on saddle point problems are presented the class of problems is generalized to include the unsymmetric case and the known stability and approximation results are strengthened and applied to the generalized problem an application to the solution of certain exterior interface problems is indicated

preconditioners for generalized saddle point problems noindent we propose and examine block diagonal preconditioners and variants of indefinite preconditioners for block two by two generalized saddle point problems that is we consider the nonsymmetric nonsingular case where the 2 2 block is small in norm and we are particularly concerned with the case where the 1 2 block is different from the transposed 2 1 block we provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices we also extend the results of de sturler and liesen siam j sci comput 26 2005 pp 1598 1619 to matrices with nonzero 2 2 block and to the use of approximate schur complements to demonstrate the effectiveness of these preconditioners we show convergence results spectra and eigenvalue bounds for two model navier stokes problems

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

finding all solutions of nonlinear equations using the dual simplex method recently an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming this algorithm is based on a simple test termed the lp test for nonexistence of a solution to a system of nonlinear equations using the dual simplex method in this letter an improved version of the lp test algorithm is proposed by numerical examples it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time

estimateur d erreur a posteriori hierarchique application aux elements finis mixtes on propose une generalisation de l estimateur a posteriori hierarchique de bank weiser aux formulation mixtes dans le cas d approximations conformes et non conformes avec ou sans integration numerique on presente comme exemples d application la formulation mixte duale du probleme de dirichlet pour le laplacien avec et sans integration numerique

a local minimax newton method for finding multiple saddle points with symmetries in this paper a local minimax newton method is developed to solve for multiple saddle points the local minimax method siam j sci comput 23 2001 pp 840 865 is used to locate an initial guess and a version of the generalized newton method is used to speed up convergence when a problem possesses a symmetry the local minimax method is invariant to the symmetry thus the symmetry can be used to greatly enhance the efficiency and stability of the local minimax method but such an invariance is sensitive to numerical error and the haar projection has been used to enforce the symmetry siam j numer anal submitted in this paper we prove that the newton method is invariant to symmetries and that such an invariance is insensitive to numerical error when a symmetric degeneracy takes place it is proved that the newton direction can be easily solved in an invariant subspace thus the newton method can be used not only to speed up convergence but also to avoid using the haar projection if the symmetric degeneracy is removable by a discretization finally numerical examples are presented to illustrate the theory

analysis of the internodes method for non conforming discretizations of elliptic equations abstract internodes is a general method to deal with non conforming discretizations of second order partial differential equations on regions partitioned into two or several subdomains it exploits two intergrid interpolation operators one for transferring the dirichlet trace across the interface the others for the neumann trace in every subdomain the original problem is discretized by the finite element method using a priori non matching grids and piece wise polynomials of different degree in this paper we provide several interpretations of the method and we carry out its stability and convergence analysis showing that internodes exhibits optimal convergence rate with respect to the finite element sizes finally we propose an efficient algorithm for the solution of the corresponding algebraic system

an augmented singular transform and its partial newton method for finding new solutions to find new solutions through the information provided by previously found solution s this paper is first to develop a new singular transform to change the local basin barrier structure of the original problem in an infinite dimensional space and then to propose an augmented partial newton method to solve the transformed problem on the solution set mathematical justifications of the new formulation and method are established details on algorithm implementation are presented examples for two very different variational pde problems are carried out to illustrate the method their numerical results are displayed by profile contour plots and error convergence data the formulation and method developed in this paper can also be applied to solve multiple fixed point problems

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

an extended hu washizu formulation for elasticity a class of new mixed formulations for elasticity is developed and analysed the formulations are based on the discrete evss method introduced in the context of incompressible viscoelastic flows by a fortin m fortin and co workers a key feature is a stabilization term that renders coercive a problem that might not otherwise be so the focus in this work is on behaviour in the incompressible limit and the goal is that of obtaining formulations that are uniformly stable and convergent concrete examples are presented of element choices that lead to unstable formulations in the classical formulation and which are stable for the formulations introduced here a selection of numerical results illustrates in a comparative way the behaviour of the elements introduced

stability of a cartesian grid projection method for zero froude number shallow water flows in this paper a godunov type projection method for computing approximate solutions of the zero froude number incompressible shallow water equations is presented it is second order accurate and locally conserves height mass and momentum to enforce the underlying divergence constraint on the velocity field the predicted numerical fluxes computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system are corrected in two steps first a mac type projection adjusts the advective velocity divergence in a second projection step additional momentum flux corrections are computed to obtain new time level cell centered velocities which satisfy another discrete version of the divergence constraint the scheme features an exact and stable second projection it is obtained by a petrov galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions the key innovation compared to existing finite volume projection methods is a correction of the in cell slopes of the momentum by the second projection the stability of the projection is proved using a generalized theory for mixed finite elements in order to do so the validity of three different inf sup conditions has to be shown the results of preliminary numerical test cases demonstrate the method s applicability on fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme

a reduced basis method for parametrized variational inequalities reduced basis methods are an efficient tool for significantly reducing the computational complexity of solving parametrized pdes originally introduced for elliptic equations they have been generalized during the last decade to various types of elliptic parabolic and hyperbolic systems in this article we extend the reduction technique to parametrized variational inequalities first we propose a reduced basis variational inequality scheme in a saddle point form and prove existence and uniqueness of the solution we state some elementary analytical properties of the scheme such as reproduction of solutions a priori stability with respect to the data and lipschitz continuity with respect to the parameters an offline online decomposition guarantees an efficient assembling of the reduced scheme which can be solved by constrained quadratic programming second we provide rigorous a posteriori error bounds with a partial offline online decomposition the reduction scheme is applied to one dimensional o

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

a stabilized mixed finite element method for elliptic systems of first order a quasilinear elliptic equation of second order can be split into a first order system in various ways we present and analyze a stabilized finite element method for the system which is well suited for any of these possible splittings under minimal assumptions on the continuous solution existence and nearly optimal convergence in l infty of the discrete solutions is established this result holds for any choice of the stabilization parameter omega 0 moreover the paper presents a framework for investigating other mixed methods for unsymmetric first order systems

finite element approximation of elliptic problems with dirac measure terms in weighted spaces applications to one and three dimensional coupled problems in this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving dirac measures using weighted sobolev spaces and weighted discrete norms our approach handles both the cases where the measure is simply a right hand side and where it represents an additional term i e solution dependent in the formulation of the problem the main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains where the coupling terms are seen as dirac measures concentrated on the fractures we first establish a decomposition lemma which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces then we consider the stability of the galerkin approximation with finite elements in weighted norms with uniform and graded meshes we introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf sup conditions in weighted norms then we focus on the convergence of the finite element method due to the lack of regularity the convergence rates are suboptimal for uniform meshes we show that using graded meshes optimal rates are recovered our theoretical results are supported by several numerical experiments finally we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three dimensional media with one dimensional fractures that are found in the analysis of microvascular flows

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

nonlinear complementarity functions for plasticity problems with frictional contact in this paper a unifying framework for the numerical treatment of frictional contact in combination with elastoplasticity with hardening is presented both kinds of problems lead to the mathematical structure of variational inequalities which are discretized by introducing additional variables for the contact stresses or the deviatoric stresses leading to a mixed formulation by defining a family of nonlinear complementarity functions which can be applied to both frictional and plastic yield conditions these inequality constraints are reformulated as equalities combined with the equilibrium equations a system of nonlinear equations is obtained that is solved in terms of a semismooth newton method to obtain a larger domain of convergence and better stability properties the introduction of additional scaling parameters is motivated and their benefit to the convergence of the scheme is demonstrated due to the locality of the contact and plastic constraints the resulting method can be interpreted as a primal dual active set strategy furthermore for linear hardening laws our method is a generalization of the well known radial return algorithm which is obtained for a special suboptimal choice of the plastic scaling parameters the efficiency and the generality of the proposed quasi newton scheme is demonstrated by several numerical examples including the simulation of a three dimensional contact problem with several elastoplastic bodies exponential isotropic hardening and coulomb friction

numerical analysis of the modified evss method abstract in this paper we present a proof of the stability of a new mixed finite element method introduced recently by the authors 9 within the context of viscoelastic fluids the mixed formulation is related to the evss elastic viscous split stress method proposed by rajagopalan et al 14 and is based on the introduction of the rate of deformation tensor as an additional unknown the proof applies to the stokes flow and some linearized constitutive equations of slow viscoelastic flows existence and uniqueness of the continuous and the discrete problems are derived from a generalized brezzi babuska theory it is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields this result allows to choose low order finite element for the stress several numerical experiments on the 4 1 contraction stokes flow will be presented which will confirm the improved stability obtained with this new formulation

a displacement based finite element formulation for incompressible and nearly incompressible cardiac mechanics the lagrange multiplier lm and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics in this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed perturbed lagrangian formulation which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of lm techniques a modified shamanskii newton raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and exploiting its equivalence modifications are developed for the penalty form focusing on accuracy we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics our results illustrate the well known influence of locking phenomena on the penalty approach particularly for lower order schemes and its effect on accuracy for whole cycle mechanics additionally we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics