Patent Application: US-81820504-A

Abstract:
the present invention relates to an estimation of computational errors appearing in the finite element calculations , particularly to method , system and program product for verification of the accuracy of numerical data measured in terms of problem - oriented criteria , where the numerical data are computed in the process of solution of a boundary value problem . in addition to the primal problem a certain adjoin problem is formed and solved . the method is based on two principles : the original and adjoint problems are solved on non - coinciding meshes , and the term presenting the product of errors arising in the primal and adjoint problems is estimated by the gradient recovery technique .

Description:
5 . technique for the error estimation in terms of problem - oriented criteria 5 . technique for the error estimation in terms of problem - oriented criteria the method presented below is designed to verify the accuracy of fe approximations , measured in terms of “ problem - oriented ” criterion that can be chosen by users . the method is applic able to various problems in mechanics and physics embracing diffusion and linear elasticity problems . it can be easily coded and attached as an independent programme - checker to the most of existing educational and industrial codes . we consider physical process that can be formally modelled in the form of a boundary value problem of elliptic type ( bvp ) as follows referring to fig1 : find a function u = u ( x 1 , . . . , x d ) of d variables x 1 , . . . , x d such that - ∑ i , j = 1 d  ∂ ∂ x i  ( a ij  ∂ u ∂ x j ) = f   in   ω , ( 1 ) ∑ i , j = 1 d  a ij  ∂ u ∂ x j  n i = g   on   γ 2 . ( 2  b ) in the above , relation ( 1 ) is a process governing equation in the solution domain ω ⊂ r d , where a ij = a ij ( x 1 , . . . , x d ), i , j = 1 , . . . , d , are the given coefficients of the problem that usually describe the diffusion properties of the respective media . this model also applies to stationary magnetic , electric , and temperature field , or some phenomena in the linear elasticity or fluid flow . the function f = f ( x 1 , . . . , x d ) can be viewed as a given source function . the coefficients are assumed to be bounded functions and function f is assumed to be square summable . further , relations ( 2a ) and ( 2b ) are boundary conditions on the boundary γ of ω , more precisely , they prescribe a behaviour of the solution and its derivatives on two nonintersecting boundary parts γ 1 and γ 2 , respectively , i . e ., γ = γ 1 ∪ γ 2 . the function u 0 = u 0 ( x 1 , . . . , x d ) belongs to the natural energy class of the problem and g = g ( x 1 , . . . , x d ) is a square summable function , symbol n i denotes the i - th component of the outward normal vector n = n ( x 1 , . . . , x d ) to the boundary γ , i . e ., n =[ n 1 , . . . , n d ] t ( see fig1 ). the above assumptions are not restrictive and cover the bulk of practically meaningful cases . let v h be a finite - dimensional space constructed by means of a selected set of finite element trial functions defined on a standard finite element mesh t h ( called the first mesh ) over the solution domain ω , see fig2 . we notice that space v h is chosen so that its functions w h vanish on γ 1 . the first approximation for problem ( 1 )-( 2 ) is defined as a function u h = u h ( x 1 , . . . , x d ) ε v h + u 0 , ∫ ω  ∑ i , j = 1 d  a ij  ∂ u h ∂ x j  ∂ w h ∂ x i   x = ∫ ω  fw h   x + ∫ γ 2  gw h   s   ∀ w h ∈ v h . ( 3 ) users are often interested not only in the overall error e = u − u h , but also in its local behaviour , e . g ., in a certain subdomain ω ⊂ ω . one way to get information about the local behaviour of u − u h is to measure the error in terms of specially selected problem - oriented criteria on “ quantity of interest ”. one of the most typical criteria of such a type is presented by the integral ∫ ω  ϕ  ( u - u h )   x , ( 4 ) where φ = φ ( x 1 . . . , x d ) is a selected weight function vanishing outside of the zone of interest . error estimators are intended to show the value of the error in the subdomain , which is the most interesting for an engineer . also they suggest information on that how to improve an approximate solution computed if the accuracy obtained is not sufficient . at this point we note that our estimator is given by the integral whose integrand could serve as the respective indicator . namely , if the accuracy achieved is not sufficient , then additional degrees of freedom ( new elements ) must be added in those parts of the solution domain , where the integrand is excessively high . b . 5 technique for the error estimation in terms of problem oriented criteria to present the technique for estimation of criterion ( 4 ), we describe two technical problems that must be previously solved . they consist of finding an approximate solution of an auxiliary problem ( so - called adjoint problem ) and making a certain post - processing of this solution v o , and also making a post - processing of the finite element approximation u h . let v τ be the second finite - dimensional space constructed by means of a selected set of finite element trial functions on another standard finite element mesh t τ ( called the second mesh ) over the solution domain ω . we notice that space v τ is chosen so that its functions w r vanish on γ , and also that t τ need not to coincide with t h , see fig2 . a part of standard finite element mesh and a patch associated with node . v τ = v τ ( x 1 , . . . , x d ) δ v τ , ∫ ω  ∑ i , j = 1 d  a ji  ∂ v τ ∂ x j  ∂ w τ ∂ x i   x = ∫ ω  ϕ   w τ   x   ∀ w τ ∈ v τ . ( 5 ) on the first mesh t h , we define the gradient averaging transformation g h mapping the gradient of the finite element approximation u h ∇ u h = [ ∂ u h ∂ x 1 , …  , ∂ u h ∂ x d ] t , ( 6 ) which is constant over each element of the finite element mesh , into a vector - valued continuous piecewise affine function g h (∇ u h )=[ g h 1 (∇ u h ), . . . , g h d (∇ u h )] t , ( 7 ) by setting each its nodal value as the mean value of ∇ u h on all elements of the patch p ( x o ) associated with corresponding node x o in the mesh t h . denote the value of i - th coordinate of gradient ∇ u h over triangle t k , let x o be one of nodes of finite element mesh t h , and let t 1 , . . . , t n xo be elements of the mesh having node x o as one of their vertices . the union of such elements is called the patch and denoted as p ( x o ). ( thus , in terms of fig2 the patch consists of six triangles , i . e ., n x = 6 .) then , we define g h i  ( ∇ u h )  ( x * ) = 1 n x *  ∑ t k ∈ p  ( x * )  ∂ u h ∂ x i   t k  , i = 1 , …  , d ,  or ( 8  a ) g h i  ( ∇ u h )  ( x * ) = 1 meast 1 + … + meast n x *  ∑ t k ∈ p  ( x * )  meast k · ∂ u h ∂ x i   t k  , i = 1 , …  , d , ( 8  b ) having g h (∇ u h ) defined at all nodes of the mesh t h , we uniquely define continuous piecewise affine function g h i (∇ u h ) over the whole domain ω . in this way , the vector - valued continuous piecewise affine function g h (∇ u h ) in ( 7 ) is built . similarly , on t τ , we define the gradient averaging transformation g τ mapping the gradient of the finite element approximation v τ ∇ v τ = [ ∂ v τ ∂ x 1 , …  , ∂ v τ ∂ x d ] t , ( 9 ) which is constant over each element of the finite element mesh , into a vector - valued continuous piecewise affine function g τ (∇ v τ )=[ g τ 1 (∇ v τ ), . . . , g τ d (∇ v τ )] t , ( 10 ) by setting each its nodal value as the mean value of ∇ v τ on all elements of the patch p ( x o ) associated with corresponding node x o in the mesh t τ . denote the value of i - th coordinate of gradient ∇ v τ over triangle t k , let x o be one of nodes of finite element mesh t τ , and let t 1 , . . . , e 0  ( u h , v τ ) = ∫ ω  f   v τ   x + ∫ γ 2  gv τ   s - ∫ ω  ∑ i , j = 1 d  a ij  ∂ u h ∂ x j  ∂ v τ ∂ x i   x ,  and ( 13 ) e 1  ( u h , v τ ) = ∫ ω  ∑ i , j = 1 d  a ij  ( ∂ u h ∂ x j - g h j  ( ∇ u h ) )  ( ∂ v τ ∂ x i - g τ i  ( ∇ v τ ) )   x . ( 14 ) t n xo be elements of the mesh having node x o as one of their vertices . the respective patch is denoted as p ( x o ). then g τ i  ( ∇ v τ )  ( x o ) = 1 n x o  ∑ t k ∈ p  ( x o )  ∂ v τ ∂ x i   t k  , i = 1 , …  , d .  or ( 11  a ) g τ i  ( ∇ v τ )  ( x o ) = 1 meast 1 + … + meast n x o  ∑ t k ∈ p  ( x o )  meast k · ∂ v τ ∂ x i   t k  , i = 1 , …  , d . ( 11  b ) having g τ i (∇ v τ )( x o ) at all nodes x o , we uniquely define continuous piecewise affine function g τ i (∇ v τ ) over the whole domain ω . in this way , the vector - valued continuous piecewise affine function g τ (∇ v τ ) in ( 10 ) is built . such averaging transformations are widely used in the finite element calculations . usually they lead to computationally inexpensive algorithms . our method estimates the quantity ( 4 ) by the quantity e ( u h , v τ ) presented by the following formula : e ( u h , v τ ):= e 0 ( u h , v τ )+ e 1 ( u h , v τ ), ( 12 ) the functional e ( u h , v τ ) is directly computable once the approximations uh and v , are found . step 1 : pose a problem of type ( 1 )-( 2a , b ); ( step a in fig1 and 18 ) step 3 : select criterion ( 4 ), i . e . subdomain ω and function φ ; ( step c ) step 4 : solve auxiliary problem ( 5 ) and find v τ ; ( step d ) step 5 : construct / form g h (∇ u h ) by ( 8a ) or ( 8b ); ( estimation step 6 : construct / form g τ (∇ v τ ) by ( 11a ) or ( 11b ); ( estimation ) step 7 : compute e 0 ( u h , v τ ); ( estimation ) step 8 : compute e 1 ( u h , v τ ); ( estimation ) we consider mechanical problem of elasticity that can be formally presented in the form of a system of linear elasticity equations as follows : find a vector - function ( called the displacement ) u = u ( x 1 , . . . , x d )=[ u 1 , . . . , u d ] t of d variables x 1 , . . . , x d such that , for i = 1 , . . . , d - ∑ j , k , l = 1 d  ∂ ∂ x j  ( l ijkl  ɛ kl  ( u ) ) = f i   in   ω , ( 1 ) ∑ j , k , l = 1 d  n j  l ijkl  ɛ kl  ( u ) = g i   on   γ 2 . ( 2  b ) in the above , relation ( 1 ) is a governing system of equations in the solution domain ( body ) ω ⊂ r d , where l ijkl = l ijkl ( x 1 , . . . , x d ), i , j = 1 , . . . , d , are the given coefficients of the problem that usually describe the elastic properties of the respective body , and the vector - function f = f ( x 1 , . . . , x d )=[ f 1 , . . . f d ] t represents a given body force . the coefficients are assumed to be bounded functions and function f is assumed to be square summable . in addition , the coefficients satisfy the following symmetry conditions further , the functions ε ij ( u ), i , j = 1 , . . . , d , are defined as follows ɛ ij  ( u ) = 1 2  ( ∂ u i ∂ x j + ∂ u j ∂ x i ) , ( 4 ) they form the so - called small strain ( or deformation ) tensor ( d × d matrix ) ε ( u ). the relations ( 2a ) and ( 2b ) are boundary conditions on the boundary γ of ω , more precisely , they prescribe a behaviour of the solution and its derivatives on two nonintersecting boundary parts γ 1 and γ 2 , respectively , i . e ., γ = γ 1 ∪ γ 2 . the vector - functions u 0 = u 0 ( x 1 , . . . , x d )=[ u 0 1 , . . . , u 0 d ] t represents the external force and belongs to natural energy class of the problem and g = g ( x 1 , . . . , x d )=[ g 1 , . . . , g d ] t is a square summable function . the function g represents the external force . the symbol n i denotes the i - th component of the outward normal vector n = n ( x 1 , . . . , x d ) to the boundary γ , i . e ., n =[ n 1 , . . . , n d ] t ( see fig3 ). above assumptions are not restrictive and cover the bulk of practically meaningful cases . let v h be a finite - dimensional space constructed by means of a selected set of finite element trial ( d - dimensional ) vector - functions defined on commonly - used finite element mesh t h over ω . we notice that space v h is chosen so that its functions w h vanish on γ 1 . the finite element approximation for problem ( 1 )-( 2 ) is defined then as a vector - function such that u h = u h ( x 1 , . . . , x d )=[ u h 1 , . . . , u h d ] t εv h + u 0 , ∫ ω  ∑ i , j , k , l = 1 d  l ijkl  ɛ ij  ( u h )  ɛ kl  ( w h )   x = ∫ ω  f · w h   x + ∫ γ 2  g · w h   s   ∀ w h ∈ v h , ( 5 ) users are often interested not only in the overall error e = u − u h , but also in its local behaviour , e . g ., in a certain subdomain ∩⊂ ω . one way to get an information about the local behaviour of u − u h is to measure the error in terms of specially selected problem - oriented criteria . one of the most typical criteria of such a type is presented by the integral ∫ ω  φ · ( u - u h )   x , ( 6 ) where φ = φ ( x 1 , . . . , x d )=[ φ 1 , . . . , φ d ] t is a selected vector - function such that φ vanishes outside of the zone of interest ( ω ). c . 4 error estimation c . 5 techniques for the error estimation in terms of problem - oriented criteria to present the techniques for estimation of criterion given by ( 6 ), we need first to describe two technical problems , that must be previously solved . they consist of finding an approximate solution of an auxiliary problem and making a certain post processing of solutions of this auxiliary problem and also of finite element approximation u h . let v r be another finite - dimensional space constructed by means of a selected set of finite element trial functions on another standard finite element mesh t τ over ω . we notice that space v τ is chosen so that its functions w τ vanish on γ , and also that t τ need not to coincide with t h . v τ = v τ ( x 1 , . . . , x d ) εv τ , ∫ ω  ∑ i , j , k , l = 1 d  l ijkl  ɛ ij  ( v τ )  ɛ kl  ( w τ )   x = ∫ ω  φ · w τ   x   ∀ w τ ∈ v τ . ( 7 ) on t h , we define the tensor averaging transformation g h mapping the tensor of the finite element approximation uh ɛ  ( u h ) = [ 1 2  ( ∂ u h i ∂ x j + ∂ u h j ∂ x i ) ] i , j = 1 d , ( 8 ) which is a constant d × d matrix over each element of the finite element mesh , into a d × d matrix - valued continuous piecewise affine function g h ( ε ( u h ))=[ g h i , j ( ε ( u h ))] i , j = 1 d , ( 9 ) by setting each its nodal value as the mean value of ε ( u h ) on all elements of the patch p ( x o ) associated with corresponding node x o in the mesh t h . more precisely , let x o be one of nodes of finite element mesh t h , and let t 1 , . . . , g h i , j  ( ɛ  ( u h ) )  ( x * ) = 1 n x *  ∑ t m ∈ p  ( x * )  1 2  ( ∂ u h i ∂ x j   t m  + ∂ u h j ∂ x i  t m ) ,  or ( 10  a ) g h i , j  ( ɛ  ( u h ) )  ( x * ) = 1 meast 1 + … + meast n x *  ∑ t m ∈ p  ( x * )  meast m · 1 2  ( ∂ u h i ∂ x j   t m  + ∂ u h j ∂ x i  t m ) . ( 10  b ) t n x . form the patch p ( x o ) ( cf . fig2 ). then , we define for i , j = 1 , . . . , d having g h i , j ( ε ( u h )) defined at all nodes of the mesh t h , we uniquely define continuous piecewise affine function g h i , j ( ε ( u h )) over the whole domain ω . in this way , the matrix - valued continuous piecewise affine function g h ( ε ( u h )) in ( 9 ) is built . similarly , on t τ , we define the tensor averaging transformation g τ mapping the tensor of the finite element approximation v r ɛ  ( v τ ) = [ 1 2  ( ∂ v τ i ∂ x j + ∂ v τ j ∂ x i ) ] i , j = 1 d , ( 11 ) which is a d × d constant matrix over each element of the finite element mesh , into a d × d matrix - valued continuous piecewise affine function g τ ( ε ( v τ ))=[ g τ i , j ( ε ( v τ ))] i , j = 1 d , ( 12 ) by setting each its nodal value as the mean value of ∇ v τ on all elements of the patch p ( x o ) associated with corresponding node x o in the mesh t τ . more precisely , let x o be one of nodes of finite element mesh t τ , and let t 1 , . . . , t n xo form the patch p ( x o ). then , we define for i , j = 1 , . . . , d g τ i , j  ( ɛ  ( v τ ) )  ( x o ) = 1 n x o  ∑ t m ∈ p  ( x o )  1 2  ( ∂ v τ i ∂ x j  t m + ∂ v τ j ∂ x i   t m )   or ( 13  a ) g τ i , j  ( ɛ  ( v τ ) )  ( x o ) = 1 meast 1 + ⋯ + meast n x o  ∑ t m ∈ p  ( x o )  meast m · 1 2  ( ∂ v τ i ∂ x j   t m  + ∂ v τ j ∂ x i   t m ) . ( 13  b ) having g r i , j ( ε ( v τ ))( x o ) at all nodes x o , we uniquely define continuous piecewise affine function g τ i , j ( ε ( v τ )) over the whole domain ω . in this way , the vector - valued continuous piecewise affine function g τ ( ε ( v τ )) in ( 12 ) is built . our method estimates the quantity ( 6 ) by the quantity e ( u h , v τ ) given by the following formula : e ( u h , v τ ):= e 0 ( u h , v τ )+ e 1 ( u h , v τ ), ( 14 ) e 0  ( u h , v τ ) = ∫ ω  f · v f   x + ∫ γ 2  g · v τ   s - ∫ ω  ∑ i , j , k , l = 1 d  l ijkl  ɛ ij  ( u h )  ɛ kl  ( v τ )   x , and ( 15 ) e 1  ( u h , v τ ) = ∫ ω  ∑ i , j , k , l = 1 d  l ijkl  ( ɛ ij  ( u h ) - g h i , j  ( ɛ  ( u h ) ) )  ( ɛ kl  ( v τ ) - g τ k , l  ( ɛ  ( v τ ) ) )   x . ( 16 ) the functional e ( u h , v τ ) is directly computable once the approximations u h and v τ are found . the algorithm is similar as that one presented in section b9 . d — advantages of the technique presented with respect to others available in the literature in the available scientific literature , several methods of computer verification of the accuracy of approximate solutions in terms of “ problem - oriented ” criteria has been presented . the main advantages of our method with respect to others can be summarized as follows : the estimator contains an extra term , e 0 , which is directly computable and contains major part of the error ; the second term is effectively estimated by means of computationally cheap procedure of “ gradient - averaging ”; the estimator is valid for a wide spectrum of approximations including the case when the meshes for the original and auxiliary ( adjoint ) problems are different . now we describe how the method can be implemented in hardware , software , firmware or combinations thereof , particularly in the form of engineering software . actually , the nature of the proposed methodology allows a development of the program block “ checker ”, fig1 , which can be easily incorporated in or added in some way to the most of the existing commercial software packages using finite element method as a main computational tool ( e . g ., matlab ®, adina ®; ansys ®, mathcad ®). to illustrate that we shall use the pde toolbox of matlab ® package , especially designed for obtaining the numerical solutions for various classes of boundary - value and initial boundary - value problems . thus , using the pde toolbox subroutines we have developed a new software , which can help the users of matlab ® not only to calculate numerical solutions for the boundary - value problems , but also effectively estimate their accuracy . naturally all necessary routines could be integrated into one software package . [ 0165 ] fig4 presents a layout of desktop of the software in a graphical user interface ( gui ) and on a display . the desktop ( control panel ) contains six main blocks : “ define problem ”— block 10 , “ mesh construction ”— block 12 , “ reference solution ”— block 14 , “ construction of adjoint problem ”— block 16 , “ estimator ”— block 18 , “ table ”— block 20 . the software works in a standard pc having processing unit , program and data memory ( ram and hard disk ), keyboard , mouse and display . the operating system is microsoft ® windows , but the software can be implemented in other operating systems , too . block 10 calls the pde toolbox main window ( fig5 ) for setting the problem : defining the solution domain , zone of interest , boundary conditions , coefficients , source function , which are exported later to our software , step a in fig1 . block 12 is used for a construction of the first mesh using mesh specification parameters of pde toolbox of matlab ® such as maximum edge size , growth rate and pde toolbox of matlab ® grid generator . the block 12 calls fe - solver of pde toolbox for finding the first finite element approximation , step b . block 14 is optional for finding the error ( brute force method ) if the user has time enough and computer resources for it . block 16 is used to construct the second mesh and find the second finite element approximation for the adjoint problem , step c . it calls pde toolbox of matlab ® grid generator and its fe - solver for refining the second mesh in the zone of interest and also globally in the whole solution domain and computing the second approximation , step d . blocks 12 , 14 , and 16 have special fields , where the parameters of corresponding meshes such as number of triangles and number of nodes are displayed . block 18 is designed to compute the estimator ( e ), step e . there is a possibility of choosing the type of gradient averaging and a parameter for calculating the integrals in the estimator ( e ), the number of triangles in the mesh used to calculate the estimator is shown in a special field . steps 5 - 9 of paragraph b . 9 form step e here . block 20 is a table designed to show the results of calculations and values of different parameters used in the processes of calculations . if user considers an error being too large , he or she can recomputed with changed initial parameters ( step f1 or f2 ). referring now to fig5 the solution domain and the zone of interest are set defining their geometry on the desktop of the pde toolbox . the coefficients and the source function are set in a known manner with the pde toolbox &# 39 ; s facilities by using its panel buttons or drop menus . the solution domain presents here a thin plate , which is heated overall by a constant heat source ( f = 10 everywhere ). the material of the plate is assumed to be homogeneous so that we can take the diagonal diffusion coefficients to be equal to 1 , and the off - diagonal coefficients to be equal to 0 . for simplicity , we assume that we have homogeneous dirichlet boundary condition only , i . e ., the temperature is assumed to be zero over the boundary of the plate . the temperature distribution in this plate can be described then by the following bvp - ∂ 2  u ∂ x 1 2 - ∂ 2  u ∂ x 2 2 = 10   in   ω , the knowledge of the temperature , especially at some critical areas , is very important for making the right design solutions , which can save , e . g ., material resources or help to improve the geometry of the plate . in the domains like we are dealing with now which have concavity in the boundary , such critical area is neighbourhood of concavity point , which dictated our choice of the zone of interest . after the setting of the problem , all the date of the problem are exported to our software , fig6 shows pde toolbox &# 39 ; s export window 26 . parameters gd refers to “ geometry data ”, sf refers to “ set formula ”, and ns refers to “ labels ” [ 0177 ] fig7 presents setting the boundary conditions using the matlab ® pde toolbox &# 39 ; s means . [ 0178 ] fig8 presents exporting the decomposed geometry and boundary conditions to checker software using the export window 26 . the export functions used in fig6 and 8 are provided by matlab ® pde toolbox . after the previously defined data have been exported to the checker software , the first finite element approximation ( primal solution ) is calculated using block 12 . the result of calculations is shown in a new popup window 22 , which presents the 3d visualization of the approximate solution . the used mesh can be obtained ( see fig9 ) by rotating the 3d visualization . block 14 by brute force method computes the reference solution on very fine mesh and tries to find the exact value of the problem - oriented criterion . this brute force method is not advisable for using in the engineering practice since it takes a large amount of computational time and memory . the brute force methods may require 100 times computer resources ( processor speed , processing time and memory ) as the new method herein described . the reference solution here is computed only for an illustration of the invention as it gives a more or less reliable reference result . the results of performance of block 14 are shown in table 20 in fig1 (“ exact error ”). ordinary user does not need block 14 . block 16 constructs the adjoint mesh and solves the adjoint problem ( with pde toolbox ). in the present implementation the function ( ù ) in the adjoint problem is set by default to 1 in the zone of interest and zero outside of it . further implementation will make it possible to set the function ( ù ) arbitrarily , which can help to analyse critical areas more carefully . fig1 shows the results of calculations of the adjoint problem in a new pop - up window 24 similarly to fig9 . in block 16 we have also a possibility to construct another adjoint mesh condensed in the zone of interest and make new calculations , which is shown in 3d visualization in fig1 . after putting window 24 to the background we can make visible the table 20 with the results of all previous calculations ( fig1 ). all the blocks compute all the approximate solutions and the estimator using formulas as described above . block 12 computes the first finite element approximation . block 16 computes the second finite element approximation ( for the adjoint / auxiliary problem ). block 18 computes the value of the estimator using the invented formula . after getting the result user is able to consider if the estimator shows that the first approximation isn &# 39 ; t enough accurate , he / she can change initial parameters and recompute a new approximation and a new estimator . the model depicts a heated rod , which is insulated but heated on its length . the rods ends are kept in zero temperature . the heating is defined by the source function . the designer gets advantage when he / she is able to compute temperature distribution more accurately and computationally cheaper in critical area . we begin with forming a simple problem : find the function u such that − u { umlaut over ()}( x )= f ( x ) in ω =( 0 , 1 ), ( 1 ) f  ( x ) = { 2  α exp  ( 1 s 2 - 1 p ) · [ - 2  ( x - x 0 ) 2 + 4  p  ( x - x 0 ) 2 + p 2 ] · 1 p 4 , if   x ∈ ( a , b ) ⋐ ( 0 , 1 ) , 0 , otherwise , p = s 2 −( x − x 0 ) 2 , s = ½ ( b − a ), x 0 = ½ ( a + b ), and α is a positive real number .) here a , b are coordinates of two sides of the rod . f ( x ) is a function that has a meaning of “ heating function ”. it shows t he amount of heat given to each elementary part of the rod . x is a coordinate changing along the rod , x 0 is a central point inside ( a , b ). s , p , and α : are positive parameters that define the shape of f ( x ). it should be noted that such a form of “ f ” is taken only for a particular example . u ( x ) has a meaning of the temperature of the rod at point x [ 0190 ] fig1 presents the shape of the source function . the exact solution of problem ( 1 )-( 2 ) is u  ( x ) = { α · exp  ( 1 s 2 - 1 p ) , if   x ∈ ( a , b ) ⋐ ( 0 , 1 ) , 0 , otherwise . the exact and approximate solutions of problem ( 1 )-( 2 ) ( full line ) and the zone of interest ( dotted line ) are marked in fig1 . ∫ 0 1  ϕ  ( x )  ( u  ( x ) - u h  ( x ) )   x ,  ϕ  ( x ) = { 1 , if   x ∈ ( z 1 , z 2 ) 0 , otherwise . ( 3 ) in what follows , m denotes the number of elements in the mesh for the primal problem , n stands for the number of elements in the mesh used for the corresponding adjoint problem . in all the tests below , m = 8 , a = 0 , b = 1 . 0 , and α = 1 . 0 . first , we present the results of numerical experiments where the uniform meshes are used , i . e ., h = 1 / m and τ = 1 / n . in this series of numerical tests , ( z 1 , z 2 )=( 0 . 375 , 0 . 625 ) ( cf . fig1 ). in table i , we present the numerical results for problem ( 1 )-( 2 ) with m = 8 and n = k · m , k = 2 , 3 , 4 , 5 , 6 , 7 , 8 . table i the results of performance of the estimator e for example 1 . m n e 0 e 1 e criterion i eff 8 16 0 . 00693883 0 . 00238174 0 . 00932058 0 . 00921705 1 . 01123239 8 24 0 . 00820668 0 . 00098981 0 . 00919649 0 . 00921705 0 . 99776967 8 32 0 . 00864915 0 . 00053743 0 . 00918658 0 . 00921705 0 . 99669427 8 40 0 . 00885372 0 . 00033653 0 . 00919025 0 . 00921705 0 . 99709278 8 48 0 . 00896479 0 . 00023026 0 . 00919505 0 . 00921705 0 . 99761349 8 56 0 . 00903174 0 . 00016737 0 . 00919911 0 . 00921705 0 . 99805336 8 64 0 . 00907518 0 . 00012710 0 . 00920229 0 . 00921705 0 . 99839820 the last column of the table presents the so - called “ effectivity index ”, which is often used to give a normalized measure of the quantity of error estimation . in our case , this index is computed as follows : i eff =  e  ( u h , v τ )   ∫ 0 1  ϕ  ( x )  ( u  ( x ) - u h  ( x ) )   x  ( 4 ) the dependence of i eff on the number k is presented on fig1 . it is easy to see that the estimator e works quite well even for very modest values of n . in this subsection , we present the results of numerical experiments , where the mesh used for solving the adjoint problem is condensed around the “ zone of interest ” ( z 1 , z 2 ). as before , h = 1 / m . now , we are aimed to show that to have a good quality of the error estimation it is not necessarily required to also use globally refined meshes for the adjoint problem . we find that essentially the same result can be obtained on the mesh refined only in the “ zone of interest ” ( i . e ., we have less degrees of freedom and , thus , save computational costs ). thus , in the tests of example 2 , the mesh t τ is only refined in a certain interval ( q 1 , q 2 ) that contains ( z 1 , z 2 ). to numerically qualify the respective meshes , we introduce positive numbers m i , i = 1 , 2 , 3 , that correspond to the number of elements in the mesh for the primal problem in the intervals ( 0 , q 1 ), ( q 1 , q 2 ), and ( q 2 , 1 . 0 ), respectively . the numbers of elements in the mesh used for the adjoint problem in the intervals ( 0 , q 1 ), ( q 1 , q 2 ), and ( q 2 , 1 . 0 ) are defined in terms of two positive numbers k 1 and k 2 as follows — m 1 · k 1 , m 2 · k 2 , and m 3 · k 1 , respectively . thus , the density of the mesh used for the adjoint problem with respect to the mesh used for the primal problem inside the interval ( q 1 , q 2 ) is given by k 2 , and outside of the interval ( q 1 , q 2 )— by k 1 . evidently , the total number of elements in the mesh used for the adjoint problem is given by n = m 1 · k 1 + m 2 · k 2 + m 3 · k 1 , see fig1 , showing meshes t h and t τ used for problem ( 1 )-( 2 ). we take ( z 1 , z 2 )=( 0 . 500 , 0 . 750 ) and ( q 1 , q 2 )=( 0 . 375 , 0 . 875 ). the results of the tests for various values of k i are presented in table ii . [ 0203 ] table ii the results of performance of the estimator e for example 2 . m 1 m 2 m 3 k 1 k 2 m n e criterion i eff 3 4 1 3 3 8 24 0 . 00495048 0 . 00500423 0 . 98925780 3 4 1 2 3 8 20 0 . 00495048 0 . 00500423 0 . 98925777 3 4 1 1 3 8 16 0 . 00495048 0 . 00500423 0 . 98925773 3 4 1 1 2 8 12 0 . 00493566 0 . 00500423 0 . 98629572 6 8 2 3 3 16 48 0 . 00122594 0 . 00122749 0 . 99873983 6 8 2 2 3 16 40 0 . 00122594 0 . 00122749 0 . 99873972 6 8 2 1 3 16 32 0 . 00122594 0 . 00122749 0 . 99873956 6 8 2 1 2 16 24 0 . 00123056 0 . 00122749 1 . 00249939 we see that the quality of the error estimation remains quite good even if what shows the effectivity of local mesh refinement in the “ zone of interest ”. referring now to fig1 to the results exposed in the table 20 we clearly observe good performance of the estimator for two dimensional problems . the “ exact error ” is based on the reference solution computed by brute force method used instead of the exact solution . the “ exact error ” means the relevant quantity of interest computed by the same formula ( 4 ) as presented paragraph b . 3 . in the above examples the models depict diffusion type processes , which arise often in various engineering applications . a typical diffusion process is heat propagation inside a body . so that physically our examples can be considered as problems on the distribution of a heat inside a body with given temperature at the boundary and a certain distribution ( f ) of the heat source . it is worth outlining that the algorithm suggested is applicable not only to such type of problems , but for all other physical models that can be described in terms of linear elliptic equations ( e . g ., linear elasticity , electrostatic problems , certain models in the theory of plates and shells ). we refer to diffusion type problems as to a simple and transparent example . 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