interpolation operators in orlicz sobolev spaces we study classical interpolation operators for finite elements like the scott zhang operator in the context of orlicz sobolev spaces furthermore we show estimates for these operators with respect to quasi norms which appear in the study of systems of p laplace type

spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

pointwise gradient estimates for a class of singular quasilinear equations with measure data abstract local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data div a x u in a bounded and possibly nonsmooth domain in r n here div a x u is modeled after the p laplacian our results extend earlier known results to the singular case in which 3 n 2 2 n 1 p 2 1 n

spline collocation method for integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions to initial or boundary value problems of fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms

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

a posteriori finite element error control for the p laplace problem three a posteriori error estimates are discussed for the p laplace problem as a model equation for nonquadratic growth of order p 2 the standard residual based error estimate is addressed with emphasis on involved constants determined as local eigenvalues up to errors in their numerical computation they provide a guaranteed upper bound for the w1 p seminorm and a weighted w1 2 seminorm of u uh under the major assumption that nabla u h is positive a second sharper a posteriori estimate is based on the numerical solution of local interface problems averaging techniques for a posteriori error control on the flux error motivate a third estimator the results are presented for any shape regular but unstructured triangulation into triangles they partly rely on a positive weight function rho nabla u p 2 nabla u h p 2 2 similar to the notion of quasi norms due to barrett and liu numerical experiments indicate the surprising accuracy of our averaging error estimators confirm our estimates and allow for a comparison

guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems abstract we derive a posteriori error estimates for a class of second order monotone quasi linear diffusion type problems approximated by piecewise affine continuous finite elements our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual as well as a global error lower bound up to a generic constant independent of the nonlinear operator they are thus fully robust with respect to the nonlinearity thanks to the choice of the error measure they are also locally efficient albeit in a different norm and hence suitable for adaptive mesh refinement moreover they allow to distinguish estimate separately and compare the discretization and linearization errors hence the iterative newton raphson fixed point linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error this can lead to important computational savings as performing an excessive number of unnecessary linearization iterations can be avoided a strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p laplacian

surmounting oscillating barriers path integral approach for weak noise we consider the thermally activated escape of an overdamped brownian particle over a potential barrier in the presence of periodic driving a time dependent path integral formalism is developed which allows us to derive asymptotically exact weak noise expressions for both the instantaneous and the time averaged escape rate our results comprise a conceptually different systematic treatment of the rate prefactor multiplying the exponentially leading arrhenius factor moreover an estimate for the deviations at finite noise strengths is provided and a supersymmetry type property of the time averaged escape rate is verified for piecewise parabolic potentials the rate expression can be evaluated in closed analytical form while in more general cases as exemplified by a cubic potential an action integral remains to be minimized numerically our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies

on the finite element approximation of p stokes systems in this paper we study the finite element approximation of systems of p stokes type for p in 1 infty we derive in some cases optimal error estimates for finite element approximation of the velocity and for the pressure in a suitable functional setting the results are supported by numerical experiments

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

higher order fem for the obstacle problem of the p laplacian a variational inequality approach abstract we consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p laplacian differential operator for p 1 we prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity moreover we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions all our results contain the known results for the linear case of p 2 we present numerical results on the improved convergence rates of adaptive schemes mesh size adaptivity with and without polynomial degree adaptation for the singular case of p 1 5 and for the degenerated case of p 3

role of an intermediate state in homogeneous nucleation we explore the role of an intermediate state phase in homogeneous nucleation by examining the decay process through a double humped potential barrier we analyze the one dimensional fokker planck fp equations with the fourth and sixth order landau potentials in the low temperature case we apply the wkb method to the fp equation and obtain an analytic expression for the decay rate which is accurate for a wide range of depth and curvature of the middle well in the case of a deep middle well it reduces to an extended kramers formula in which the barrier height in the original formula is replaced by the arithmetic mean height of the higher outer and lower inner barriers and the curvature of the initial well in the original one is replaced by the geometric mean curvature of the initial and intermediate wells in the case of a shallow middle well the kramers escape rate is evaluated also within the standard framework of the mean first passage time problem whose result is consistent with our wkb analysis criteria for enhancement of the decay rate are revealed

smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems abstract we construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form d y t f t y t with caputo type fractional derivatives d y of order 0 using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed algorithms two numerical examples are given

convergence of adaptive finite element methods for general second order linear elliptic pdes we prove convergence of adaptive finite element methods afems for general nonsymmetric second order linear elliptic pdes thereby extending the result of morin nochetto and siebert it siam j numer anal 38 2000 pp 466 488 it siam rev 44 2002 pp 631 658 the proof relies on quasi orthogonality which accounts for the bilinear form not being a scalar product together with novel error and oscillation reduction estimates which now do not decouple we show that afems are a contraction for the sum of energy error plus oscillation numerical experiments including oscillatory coefficients and both coercive and noncoercive convection diffusion pde illustrate the theory and yield optimal meshes

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

analysis of an adaptive uzawa finite element method for the nonlinear stokes problem we design and study an adaptive algorithm for the numerical solution of the stationary nonlinear stokes problem the algorithm can be interpreted as a disturbed steepest descent method which generalizes uzawa s method to the nonlinear case the outer iteration for the pressure is a descent method with fixed step size the inner iteration for the velocity consists of an approximate solution of a nonlinear laplace equation which is realized with adaptive linear finite elements the descent direction is motivated by the quasi norm which naturally arises as distance between velocities we establish the convergence of the algorithm within the framework of descent direction methods

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

numerical simulations of 2d fractional subdiffusion problems the growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes currently most algorithms are designed for 1d problems due to the memory effect in fractional derivatives in this work the 2d fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach the proposed algorithm is also used to simulate a subdiffusion convection equation

nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient nonlinear multigrid methods such as the full approximation scheme fas and newton multigrid newton mg are well established as fast solvers for nonlinear pdes of elliptic and parabolic type in this paper we consider newton mg and fas iterations applied to second order differential operators with nonlinear diffusion coefficient under mild assumptions arising in practical applications an approximation shown to be sharp of the execution time of the algorithms is derived which demonstrates that newton mg can be expected to be a faster iteration than a standard fas iteration for a finite element discretisation results are provided for elliptic and parabolic problems demonstrating a faster execution time as well as greater stability of the newton mg iteration results are explained using current theory for the convergence of multigrid methods giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice

implicit runge kutta methods of optimal order for volterra integro differential equations implicit runge kutta methods with m stages and optimal order p 2m for the approximate solution of volterra integro differential equations can be viewed as fully dis cretized collocation methods in certain polynomial spline spaces the choice of the quadrature formulas needed for the full discretization of the collocations is investigated and it is shown that in contrast to ordinary differential equations there exist for fixed m several optimal methods 1 introduction in this paper we shall be concerned with the approximate solution of the initial value problem for a nonlinear first order volterra integro differential equation

a block by block method for volterra integro differential equations with weakly singular kernel the theory of a block by block method for solving volterra integro differential equations with continuous kernels see makroglou 14 5d is adapted to volterra integrodifferential equations with weakly singular kernels and a rate of convergence is given

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

global regularity properties of steady shear thinning flows abstract in this paper we study the regularity of weak solutions to systems of p stokes type describing the motion of shear thinning fluids in certain steady regimes in particular we address the problem of regularity up to the boundary improving previous results especially in terms of the allowed range for the parameter p

linear convergence of an adaptive finite element method for the p laplacian equation we study an adaptive finite element method for the p laplacian like pdes using piecewise linear continuous functions the error is measured by means of the quasi norm of barrett and liu we provide residual based error estimators without a gap between the upper and lower bound we show linear convergence of the algorithm which is similar to the one of morin nochetto and siebert all results are obtained without extra marking for the oscillation

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

higher order calderon zygmund estimates for the p laplace equation abstract the paper is concerned with higher order calderon zygmund estimates for the p laplace equation div a u div u p 2 u div f 1 p we are able to transfer local interior besov and triebel lizorkin regularity up to first order derivatives from the force term f to the flux a u for p 2 we show that f b q s implies a u b q s for any s 0 1 and all reasonable q 0 in the planar case the result fails for p 2 in case of higher dimensions and systems we have a smallness restriction on s the quasi banach case 0 min q 1 is included since it has important applications in the adaptive finite element analysis as an intermediate step we prove new linear decay estimates for p harmonic functions in the plane for the full range 1 p

piecewise polynomial collocation for fredholm integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of initial or boundary value problems of linear fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

finite element approximation of the p laplacian in this paper we consider the continuous piecewise linear finite el ement approximation of the following problem given p 1 oo and g find u such that v vu 2vu f iniicr2 u g on a the finite element approximation is defined over i2 a union of regular tri angles yielding a polygonal approximation to q for sufficiently regular so lutions u achievable for a subclass of data g and i2 we prove optimal error bounds for this approximation in the norm wlq q 1 q p for p 2 under the additional assumption that qh c 2 numerical results demonstrating these bounds are also presented

dynamics of gompertzian tumour growth under environmental fluctuations abstract we investigate the effect of correlated additive and multiplicative gaussian white noise on the gompertzian growth of tumours our results are obtained by solving numerically the time dependent fokker planck equation fpe associated with the stochastic dynamics in our numerical approach we have adopted b spline functions as a truncated basis to expand the approximated eigenfunctions the eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study we perform simulations to analyze various aspects of the probability distribution of the tumour cell populations in the transient and steady state regimes more precisely we are concerned mainly with the behaviour of the relaxation time to the steady state distribution as a function of i of the correlation strength between the additive noise and the multiplicative noise and ii as a function of the multiplicative noise intensity d and additive noise intensity it is observed that both the correlation strength and the intensities of additive and multiplicative noise affect the relaxation time

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

a variational approach to stochastic nonlinear problems a variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a bernoulli oscillator driven by white noise comparison with exact data available for this system shows that the variational approach to such problems can be quite effective

a posteriori fe error control for p laplacian by gradient recovery in quasi norm a posteriori error estimators based on quasi norm gradient recovery are established for the finite element approximation of the p laplacian on unstructured meshes the new a posteriori error estimators provide both upper and lower bounds in the quasi norm for the discretization error the main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi norm

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

quasi norm interpolation error estimates for the piecewise linear finite element approximation of p laplacian problems in this work new interpolation error estimates have been derived for some well known interpolators in the quasi norms the estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations in particular by using these estimates we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2 d and 3 d p laplacian

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

shifted jacobi spectral galerkin method for solving fractional order initial value problems abstract in this paper the shifted jacobi spectral galerkin method is introduced to deal with fractional order initial value problems fivps in the proposed method the exact solution of the fivp is approximated by using shifted jacobi polynomials on each subinterval of the total time the main advantage of the proposed method is that the rate of convergence in l 2 norm depends on the local smoothness of solution this enables the proposed method to work for nonsmooth solutions the second merit is that the flexibility between the length of sub interval and the degree of polynomial significantly enhances the numerical accuracy we derive the spectral galerkin approximation formula for the volterra integral equation of the second kind which is equivalent to the fivp error analysis in l 2 norm for the case 0 in shifted jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is lipschitz continuous in the second argument numerical illustrations for multi order linear and nonlinear fivps are demonstrated to confirm the convergence rate

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

on the convergence of adaptive nonconforming finite element methods for a class of convex variational problems we formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems the class of minimization problems we admit includes highly singular problems for which no euler lagrange equation or inequality is available as a consequence our arguments only use the structure of the energy functional we are nevertheless able to prove convergence of an adaptive algorithm using even refinement indicators that are not reliable error indicators

post processing discontinuous galerkin solutions to volterra integro differential equations analysis and simulations this paper presents a superconvergence extraction technique for volterra integro differential equations with smooth and non smooth kernels specifically extracting superconvergence is done via a post processed discontinuous galerkin dg method obtained from interpolating the dg solution using lagrange polynomials at the nodal points a global superconvergence error bound in the l norm is established for a non smooth kernel a family of non uniform time meshes is used to compensate for the singular behaviour of the exact solution near t 0 the derived theoretical results are numerically validated in a sample of test problems demonstrating higher than expected convergence rates

approximation methods for second kind weakly singular volterra integral equations abstract projection methods are applied to obtain the convergence rates for volterra integral equations with weakly singular kernels we consider galerkin and multi galerkin methods and their iterated versions to solve volterra integral equations with weakly singular kernels in the space of piecewise polynomials subspaces based on graded mesh we will show that the iterated multi galerkin method improves over iterated galerkin method in fact we show that iterated multi galerkin solution converges with the convergence rates o n 3 m and o n 3 m log n 2 for algebraic and logarithmic type kernels respectively we prove that iterated galerkin method for algebraic kernel converges with the convergence rate o n 2 m and for logarithmic type kernel converges with the convergence rate o n 2 m log n where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations theoretical results are justified by the numerical results

diffusion in a bistable potential the functional integral approach we study with the help of the onsager machlup functional integral approach the distributionp of a single stochastic variable the evolution of which is described by a fokker planck equation with a first moment deriving from a bistable potential we set up the approximation scheme appropriate in this approach to the limit of constant and small diffusion coefficient two regimes are to be distinguished very long times kramers regime are treated within the frame of a free instanton molecule gas approximation and at intermediate times suzuki regime a standard semiclassical calculation is legitimate we thus rederive exactly the results obtained from the mode expansion and wkb method

hp discontinuous galerkin time stepping for volterra integrodifferential equations we present an hp error analysis of the discontinuous galerkin time stepping method for volterra integrodifferential equations with weakly singular kernels we derive new error bounds that are explicit in the time steps the degrees of the approximating polynomials and the regularity properties of the exact solution it is then shown that start up singularities can be resolved at exponential rates of convergence by using geometrically graded time steps our theoretical results are confirmed in a series of numerical tests

analysis of an adaptive finite element method for recovering the robin coefficient based on a new a posteriori error estimator an adaptive finite element method is proposed for recovering the robin coefficient involved in a diffusion system from some boundary measurement the a posteriori error estimator cannot be derived for this ill posed nonlinear inverse problem as was done for the existing a posteriori error estimators for direct problems instead we shall derive the a posteriori error estimator from our convergence analysis of the adaptive algorithm we prove that the adaptive algorithm guarantees a convergent subsequence of discrete solutions in an energy norm to some exact triplet the robin coefficient state and costate variables determined by the optimality system of the least squares formulation with tikhonov regularization for the concerned inverse problem some numerical results are also reported to illustrate the performance of the algorithm

diffusion in a potential field path integral approach abstract within the framework of the path integral formalism we develop a simple method to solve the fokker planck diffusion problems in a potential field including the decay of an unstable state unlike previous approaches it yields a unified treatment of all time regimes and is legitimate for any value of the diffusion coefficient an attractive feature of our method is that it provides numerical results in regions where there are no analytical solutions in particular being valid for an arbitrary shape of the potential it is very useful for studying a diffusion problem near and in the critical point where the standard treatments break down moreover our method after a simple transformation permits to obtain also the first passage times of the process

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

a superconvergent discontinuous galerkin method for volterra integro differential equations smooth and non smooth kernels abstract we study the numerical solution for volerra integro differential equations with smooth and non smooth kernels we use an h version discontinuous galerkin dg method and derive nodal error bounds that are explicit in the parameters of interest in the case of non smooth kernel it is justified that the start up singularities can be resolved at superconvergence rates by using non uniformly graded meshes our theoretical results are numerically validated in a sample of test problems

numerical estimation of the path integral and the diffusion in an anharmonic oscillator abstract the solution of the fokker planck equation given in the form of the path integral is estimated by the contribution from paths around the most probable path a numerical procedure for this estimation is given and applied to the regression of an anharmonic oscillator under a random force from unstable positions to a stable position

decay rates of adaptive finite elements with dorfler marking we investigate the decay rate for an adaptive finite element discretization of a second order linear symmetric elliptic pde we allow for any kind of estimator that is locally equivalent to the standard residual estimator this includes in particular hierarchical estimators estimators based on the solution of local problems estimators based on local averaging equilibrated residual estimators the zz estimator etc the adaptive method selects elements for refinement with dorfler marking and performs a minimal refinement in that no interior node property is needed based on the local equivalence to the residual estimator we prove an error reduction property in combination with minimal dorfler marking this yields an optimal decay rate in terms of degrees of freedom

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a relaxed kacanov iteration for the p poisson problem in this paper we introduce and analyze an iteratively re weighted algorithm that allows to approximate the weak solution of the p poisson problem for 1 p leqslant 2 by iteratively solving a sequence of linear elliptic problems the algorithm can be interpreted as a relaxed kacanov iteration as so called in the specific literature of the numerical solution of quasi linear equations the main contribution of the paper is proving that the algorithm converges at least with an algebraic rate

quasi norm local error estimators for p laplacian in this paper we extend the quasi norm techniques used in a priori error estimation of finite element approximation of degenerate nonlinear systems in order to carry out an improved a posteriori error analysis for the p laplacian we derive quasi norm a posteriori error estimators of residual type which are shown to provide improved upper and lower bounds on the discretization error for sufficiently regular solutions these estimators are further shown to be equivalent on the discretization error in a quasi norm numerical results demonstrating these a posteriori estimators are also presented

axioms of adaptivity this paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of discrete lower bounds inexact solvers inhomogeneous boundary data or the use of equivalent error estimators solely four axioms guarantee the optimality in terms of the error estimators compared to the state of the art in the temporary literature the improvements of this article can be summarized as follows first a general framework is presented which covers the existing literature on optimality of adaptive schemes the abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method second efficiency of the error estimator is neither needed to prove convergence nor quasi optimal convergence behavior of the error estimator in this paper efficiency exclusively characterizes the approximation classes involved in terms of the best approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant third some general quasi galerkin orthogonality is not only sufficient but also necessary for the r linear convergence of the error estimator which is a fundamental ingredient in the current quasi optimality analysis due to stevenson 2007 finally the general analysis allows for equivalent error estimators and inexact solvers as well as different non homogeneous and mixed boundary conditions

on quasi norm interpolation error estimation and a posteriori error estimates for p laplacian in this paper we establish a series of interpolation error estimates for several widely used averaging interpolators in some quasi norms these estimates are among the key ingredients in our improved a posteriori error analysis for the p laplacian the quasi norm interpolation error estimates are used to derive a posteriori error estimators for some conforming and nonconforming finite element approximation of the p laplacian which are shown to provide upper and lower bounds for the discretization error

decay to the nonequilibrium steady state of the thermal diffusion in a tilted periodic potential abstract we investigate the phenomenon of asymptotic decay towards the nonequilibrium steady state of the thermal diffusion in the presence of a tilted periodic potential the parameter dependence of the decay rate is revealed by investigating the fokker planck fp equation in the low temperature case under the spatially periodic boundary condition pbc we apply the wkb method to the associated schrodinger equation while eigenvalues of the non hermitian fp operator are complex in general in a small tilting case the imaginary parts of the eigenvalues are almost vanishing then the schrodinger equation is solved with pbc the decay rate is analyzed in the context of quantum tunneling through a triple well effective periodic potential in a large tilting case the imaginary parts of the eigenvalues of the fp operator are crucial we apply the complex valued wkb method to the schrodinger equation with the absorbing boundary condition finding that the decay rate saturates and depends only on the temperature the potential periodicity and the viscous constant the intermediate tilting case is also explored the analytic results agree well with the numerical data for a wide range of tilting