on the construction of deflation based preconditioners in this article we introduce new bounds on the effective condition number of deflated and preconditioned deflated symmetric positive definite linear systems for the case of a subdomain deflation such as that of nicolaides siam j numer anal 24 1987 pp 355 365 these theorems can provide direction in choosing a proper decomposition into subdomains if grid refinement is performed keeping the subdomain grid resolution fixed the condition number is insensitive to the grid size subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication numerical experiments for a steady state convection diffusion problem are included

black box multigrid for nonsymmetric problems one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

the frequency decomposition multi grid method a new variant of the multi grid algorithms is presented it uses multiple coarse grid corrections with particularly associated prolongations and restrictions in this paper the robustness with respect to anisotropic problems is considered

numerical method for solving matrix coefficient elliptic equation with sharp edged interfaces solving elliptic equations with sharp edged interfaces is a challenging problem for most existing methods especially when the solution is highly oscillatory nonetheless it has wide applications in engineering and science an accurate and efficient method is desired we propose an efficient non traditional finite element method with non body fitting grids to solve the matrix coefficient elliptic equations with sharp edged interfaces extensive numerical experiments show that this method is second order accurate in the l norm and that it can handle both sharp edged interface and oscillatory solutions

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

a new approach to the wave equation an application of the decomposition method abstract in the present paper the inhomogeneous wave equation is solved quickly easily and elegantly by adomian s decomposition method

piecewise polynomial discretization and krylov accelerated multigrid for elliptic interface problems a new piecewise polynomial interface method pim for discretizing elliptic problems with complex interfaces between high contrast materials is derived analyzed and tested a krylov accelerated interface multigrid approach img solves the discretization efficiently stability and convergence are proved in one dimension while an extensive array of numerical experiments with complex interfaces and large coefficient transitions demonstrate the accuracy efficiency and robustness of the method in two dimensions

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

level set methods for the simulation of epitaxial phenomena we introduce a model for epitaxial phenomena based on the motion of island boundaries which is described by the level set method our model treats the growing film as a continuum in the lateral direction but retains atomistic discreteness in the growth direction an example of such an island dynamics model using the level set method is presented and compared with the corresponding rate equation description extensions of our methodology to more general settings are then discussed s1063 651x 98 50212 7

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

the decomposition approach to inverse heat conduction adomian s decomposition approach is employed for solving some inverse boundary value problems in heat conduction further the mollification method is applied to deal with noisy input data and obtain a stable approximate solution

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

solving frontier problems modelled by nonlinear partial differential equations abstract we present a further development of the decomposition method 1 2 which leads to a single efficient and global method of solving linear or nonlinear ordinary or partial differential equations for initial value or boundary value problems no linearization perturbation or resort to discretized methods is involved potential savings in computation are very large perhaps six orders of magnitude in some cases and important implications exists for modeling and computational analysis it is to be noted that once we realize that we can be less constrained by the mathematics by removing the necessity of techniques such as linearization perturbation or discretization to make analysis and computation of the models feasible and practical we become able to develop more sophisticated and realistic models modeling effectively means retention of essential features while striving for simplicity so that the resulting equations can be solved with fewer limitations imposed to achieve tractibility the models can be more realistic and we have a convinient and global technique for solution

global existence singularities and ill posedness for a nonlocal flux in this paper we study a one dimensional model equation with a nonlocal flux given by the hilbert transform that is related with the complex inviscid burgers equation this equation arises in different contexts to characterize nonlocal and nonlinear behaviors we show global existence local existence blow up in finite time and ill posedness depending on the sign of the initial data for classical solutions

solution of nonlinear vibration problem of a prestressed membrane by adomian decomposition aj adomian polynomial for the jth ordinary differential equation e modulus of elasticity gj part of the jth equation independent of the dependent variables lj easily invertible operator in the jth ordinary differential equation nj nonlinear operator in the jth ordinary differential equation rj remaining linear part in the jth ordinary differential equation u displacement along the x direction v displacement along the y direction w displacement along the z direction poisson s ratio density

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

on the well posedness of various one dimensional model equations for fluid motion abstract we consider 1d equations with nonlocal velocity of the form w t u w x u x w w where the nonlocal velocity u is given by 1 u 1 x x w 0 or 2 u h w h is the hilbert transform in this paper we address several local well posedness results with blow up criteria for smooth initial data we then establish the global well posedness by using the blow up criteria

a geometric multigrid solver for the free surface equation in environmental models featuring irregular coastlines a recently developed multigrid method botto 2013 based on the concept of volume fraction has been tested for the inversion of the helmholz type equation for the free surface in the environmental public domain code coherens the volume fraction concept is particularly interesting for coarse grid cells that are agglomerated from both dry and wet fine grid cells at irregular coastlines at these locations modifying the prolongation operator and the coarse grid discretization using the volume fraction results in better convergence however as convergence deteriorates in the case of small elongated islands that tend to disappear by the multigrid coarsening procedure a correction is proposed yielding good convergence rates irrespective of the presence of small or large islands the method is tested extensively for the inversion of the academic poisson equation larger test cases solving the helmholz type equation prove the applicability for real life applications of environmental flows

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

the noisy convergence phenomena in decomposition method solutions abstract a damped oscillating convergence to a correct solution sometimes observed in decomposition method solutions of differential and partial differential equation is investigated

a computational algebraic investigation of the decomposition method for time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat or wave equations is investigated numerically using computational algebraic techniques

a simple weak formulation for solving two dimensional diffusion equation with local reaction on the interface abstract in this paper we propose a numerical method for solving two dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface we use finite element method coupled with newton s method to deal with the jump conditions and to linearize the system it is easy to implement the grid used here is body fitting grids based on the idea of semi cartesian grid numerical experiments show that this method is nearly second order accurate in the l norm

finite difference ghost point multigrid methods on cartesian grids for elliptic problems in arbitrary domains in this paper we present a numerical method for solving elliptic equations in an arbitrary domain described by a level set function with general boundary conditions dirichlet neumann robin etc on cartesian grids using finite difference discretization and non eliminated ghost values a system of n i n g equations in n i n g unknowns is obtained by finite difference discretization on the n i internal grid points and second order interpolation to define the conditions for the n g ghost values the resulting large sparse linear system is then solved by a multigrid technique the novelty of the papers can be summarized as follows general strategy to discretize the boundary condition to second order both in the solution and its gradient a relaxation of inner equations and boundary conditions by a fictitious time method inspired by the stability conditions related to the associated time dependent problem with a convergence proof for the first order scheme an effective geometric multigrid which maintains the structure of the discrete system at all grid levels it is shown that by increasing the relaxation step of the equations associated to the boundary conditions a convergence factor close to the optimal one is obtained several numerical tests including variable coefficients anisotropic elliptic equations and domains with kinks show the robustness efficiency and accuracy of the approach

a weak formulation for solving elliptic interface problems without body fitted grid a typical elliptic interface problem is casted as piecewise defined elliptic partial differential equations pde in different regions which are coupled together with interface conditions such as jumps in solution and flux across the interface in many situations such as the interface is moving the challenge is how to solve such a problem accurately robustly and efficiently without generating a body fitted mesh the key issue is how to capture complex geometry of the interface and jump conditions across the interface effectively on a fixed mesh while the interface is not aligned with the mesh and the pde is not valid across the interface in this work we present a systematic formulation and further study of a second order accurate numerical method proposed in hou and liu 2005 16 for elliptic interface problem the key idea is to decompose the solution into two parts a singular part and a regular part the singular part captures the interface conditions while the regular part belongs to an appropriate space in the whole domain which can be solved by a standard finite element formulation in a general setup the two parts are coupled together we give an explicit study of the construction of the singular part and the discretized system for the regular part one key advantage of using weak formulation is that one can avoid assuming using more regularity than necessary of the solution and the interface we present the numerical algorithm and numerical tests in 3d to demonstrate the accuracy and other properties of our method

blow up of solutions for a 1d transport equation with nonlocal velocity and supercritical dissipation we study a 1d transport equation with nonlocal velocity and supercritical dissipation we show that for a certain class of initial data the solution blows up in finite time

derivation of the adomian decomposition method using the homotopy analysis method abstract adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems it has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits however very little has been done to address the mathematical foundation of the method and its error analysis in this article the mathematical derivation of the method using the homotopy analysis method is presented in addition an error analysis is addressed as well as the convergence criteria

numerical study of modified adomian s method applied to burgers equation the application of adomian s decomposition method to partial differential equations when the exact solution is not reached demands the use of truncated series but the solution s series may have small convergence radius and the truncated series may be inaccurate in many regions in order to enlarge the convergence domain of the truncated series pade approximants pas to the adomian s series solution have been tested and applied to partial and ordinary differential equations with good results in this paper pas both in x and t directions applied to the truncated series solution given by adomian s decomposition technique for burgers equation are tested numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution it is also shown in this paper that the application of adomian s method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences the so called method of lines may reduce the convergence domain of the solution s series

an energy minimizing interpolation for robust multigrid methods we propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation it can handle pde coefficients of various types on structured or unstructured grids under one framework the formulation is general it can be applied to any dimension we demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem an oscillatory coefficient problem and an anisotropic problem empirically the convergence rate is independent of the coefficients of the underlying pde in addition to being independent of the mesh size the proposed method is primarily designed for second order elliptic pdes with possible extensions to other classes of problems such as integral equations

the decomposition method for forward and backward time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat wave and beam equations is investigated for both forward direct and backward inverse problems it is shown that for forward problems the convergence is faster than for backward problems

well posedness for a transport equation with nonlocal velocity abstract we study a one dimensional transport equation with nonlocal velocity which was recently considered in the work of cordoba cordoba and fontelos a cordoba d cordoba m a fontelos formation of singularities for a transport equation with nonlocal velocity ann of math 2 162 3 2005 1377 1389 we show that in the subcritical and critical cases the problem is globally well posed with arbitrary initial data in h max 3 2 0 while in the supercritical case the problem is locally well posed with initial data in h 3 2 and is globally well posed under a smallness assumption some polynomial in time decay estimates are also discussed these results improve some previous results in a cordoba d cordoba m a fontelos formation of singularities for a transport equation with nonlocal velocity ann of math 2 162 3 2005 1377 1389

the decomposition method for initial value problems abstract in this paper initial value problems for the heat wave and plate equations in which the vectorial space variable varies in the whole r d will be handled using the reliable decomposition method the solutions are obtained in the form of rapidly convergent power series with elegantly computable terms showing that the new technique is reliable powerful and promising

a simple method for matrix valued coefficient elliptic equations with sharp edged interfaces abstract the traditional finite element method has a number of nice properties and thus it is highly desired for matrix valued coefficient elliptic equations with sharp edged interfaces however its efficient implementation with body fitting grids for such problems is highly nontrivial in this paper we propose a simple finite element method with body fitting grids based on semi cartesian grid which makes its implementation fairly straightforward even for complicated geometry all possible situations that the interface cuts the grid are considered further the symmetry and positive definiteness of the resulting matrix are ensured for positive definite coefficients numerical experiments show that it is second order accurate in the l norm and numerically very stable

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

second order finite difference ghost point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface abstract in this paper we propose a second order accurate numerical method to solve elliptic problems with discontinuous coefficients with general non homogeneous jumps in the solution and its gradient in 2d and 3d the method consists of a finite difference method on a cartesian grid in which complex geometries boundaries and interfaces are embedded and is second order accurate in the solution and the gradient itself in order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface two numerical values are assigned on grid points that are close to the interface a real value that represents the numerical solution on that grid point and a ghost value that represents the numerical solution extrapolated from the other side of the interface obtained by enforcing the assigned non homogeneous jump conditions on the solution and its flux the method is also extended to the case of matrix coefficient the linear system arising from the discretization is solved by an efficient multigrid approach unlike the 1d case grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient a proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor comparable with the one obtained by local fourier analysis for rectangular domains the method is robust enough to handle large jump in the coefficients order of accuracy monotonicity of the errors and good convergence factor are maintained by the scheme

the maximum principle and the global attractor for the dissipative 2d quasi geostrophic equations the long time behavior of the solutions to the two dimensional dissipative quasi geostrophic equations is studied we obtain a new positivity lemma which improves a previous version of a cordoba and d cordoba 10 and 11 as an application of the new positivity lemma we obtain the new maximum principle i e the decay of the solution in l p for any p 2 when f is zero as a second application of the new positivity lemma for the sub critical dissipative case with open image in new window the existence of the global attractor for the solutions in the space h s for any s 2 1 is proved for the case when the time independent f is non zero therefore the global attractor is infinitely smooth if f is this significantly improves the previous result of berselli 2 which proves the existence of an attractor in some weak sense for the case 1 the global attractor exists in h s for any s 0 and the estimate of the hausdorff and fractal dimensions of the global attractor is also available

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

a homotopy perturbation technique for solving partial differential equations of fractional order in finite domains abstract in this paper a homotopy perturbation technique is proposed to solve a class of initial boundary value problems of partial differential equations of arbitrary fractional orders over finite domains the basic idea of this technique is to utilize both the initial and boundary conditions in the recursive relation of the solution scheme so that we can obtain a good approximate solution numerical examples are presented to illustrate the validity of the proposed technique

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods

a numerical method for solving variable coefficient elliptic equation with interfaces a new 2nd order accurate numerical method on non body fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains w separated by interfaces c the variable coefficients the source term and hence the solution itself and its derivatives may be discontinuous across the interfaces jump conditions in solution and its co normal derivative at interface are prescribed instead of smooth the interfaces are only required to be lipschitz continuous as submanifold a weak formulation is developed the existence uniqueness and regularity of the solutions are studied the numerical method is derived by discretizing the weak formulation the method is different from traditional finite element methods extensive numerical experiments are presented and show that the method is 2nd order accurate in solution and 1st order accurate in its gradient in l norm if the interface is c 2 and solutions are c 2 on the closures of the subdomains the method can handle the problems when the solutions and or the interfaces are weaker than c 2 for example email protected h 2 w c is lipschitz continuous and their singularities coincide see example 18 in section 4 the accuracies of the method under various circumstances are listed in table 19

the polynomial trefftz method for solving backward and inverse source wave problems the trefftz method is a truly meshless boundary type method because the trial solutions automatically satisfy the governing equation in order to stably solve the high dimensional backward wave problems and the one dimensional inverse source problems we develop a multiple scale polynomial trefftz method msptm of which the scales are determined a priori by the collocation points the msptm can retrieve the missing initial data and unknown time varying wave source the present method can also be extended to solve the higher dimensional wave equations long term through the introduction of a director in the two dimensional polynomial trefftz bases several numerical examples reveal that the msptm is efficient and stable for solving severely ill posed inverse problems of wave equations under large noises