ALF USER GUIDE Contents 1. Introduction 1 2. User interface 2 3
Contents
1. _ Draw _ QUIT FIGURE 2 ALF s graphics window You can compute a new set of examples by adjusting the corre sponding choices and text fields in the main window and clicking the Compute button Upon clicking the button in the main window you ll find in the output area a short user guide for ALF Clicking the Quit button terminates ALF 3 DOMAIN AND INITIAL GRID The choice Domain and initial grid offers the following options for choosing the domain of the partial differential equation and the coarsest partition for the finite element discretization e Square 8 triangles The domain is the square 1 1 x 1 1 The initial partition consists of eight isosceles right angled tri angles with short sides of length 1 and longest side parallel to the line y z R VERFURTH e Square 4 triangles The domain is the square 1 1 x 1 1 The initial partition is obtained by drawing the two diagonals of the square e Square 4 squares The domain is the square 1 1 x 1 1 The initial partition consists of four squares with sides of length 1 e L shaped 6 triangles The domain and initial partition are ob tained by removing the square 0 1 x 1 0 in the example Square 8 triangles e L shaped 3 squares The domain and initial partition are ob tained by removing the square 0 1 x 1 0 in the example Square 4 squares e Circle 8 triangles The domain is the2. FANAT ANT A PIX SOY oe level 4 Draw Quit FIGURE 4 Partition resulting from 4 steps of uniform refinement applied to the grid of Fig 3 __ ALF Graphics first level E ori E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 Finest level 4 Draw Quit FIGURE 5 Initial partition of the example Circle 4 squares 8 triangles corresponding elements When refining an element having an edge with its endpoints on the circle s boundary the midpoint of the edge will be projected onto the circle Segment 3 squares 6 triangles The domain and initial parti tion are obtained from those of the example Circle 4 squares 8 triangles by removing the quarter segment in the quadrant x gt 0 y lt 0 and the corresponding elements When refining an element having an edge with its endpoints on the circle s boundary the midpoint of the edge will be projected onto the circle R VERFURTH e Annulus 12 triangles The domain is the annulus which is formed by concentric circles with radii 0 25 and 1 and centre at the origin Figures 6 and 7 show for this example the initial partition and the mesh that is obtained after 4 steps of uniform refinement When refining an element having an edge with its endpoints on the boundary the midpoint of the edge will be projected onto the corresponding circle 6000 ALF Graphics first level F gr
3. Indicates an erroneous user input when using the options user defined initial grid or user defined boundary automatic tessellation e Inconsistent element data Indicates an erroneous user input when using the options user defined initial grid or user de fined boundary automatic tessellation e No neighbour found for element with index at least In dicates an erroneous user input when using the options user defined initial grid or user defined boundary automatic tes sellation e Too many nodes in first grid Should not appear e Too many elements in first grid Should not appear e Non existing initial grid Should not appear e No elements in first grid Should not appear When using a too large number of refinement levels or more critical a too large number of nodes your Java virtual machine may run out of memory It then throws an out of memory exception ALF tries to catch this exception In this case ALF prints the error message Error insufficient memory You then have to re launch the applet If the error message did occur with your first example you probably have chosen a too large a number of nodes In this case you should enter a smaller number in the text field Maximum number of nodes and try again If the error message occurs after several successful computations Java s garbage collector probably has not done its job In this case you may try again with your old
4. Non homogeneous Neumann boundary data g give rise to an additional boundary integral on the right hand side of the discrete problem The choice Discretization provides two options for building the discrete problem e With numerical quadrature All integrals are evaluated using a midpoint rule e Without numerical quadrature All integrals are evaluated ex actly subject to the assumption that all functions u v D a b f etc are piecewise linear respectively bilinear 6 SOLVER The choice Solver allows you to choose among the following solution algorithms for the discrete problems e MG V cycle 1 2 symmetric Gauss Seidel step One iteration consists of one multigrid V cycle with one forward Gauss Seidel sweep for pre smoothing and one backward Gauss Seidel sweep for post smoothing e MG V cycle n symmetric Gauss Seidel steps One iteration consists of one multigrid V cycle with a user specified number of symmetric Gauss Seidel sweeps for pre and post smoothing You will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps ALF USER GUIDE 13 e MG W cycle 1 2 symmetric Gauss Seidel step One iteration consists of one multigrid W cycle with one forward Gauss Seidel sweep for pre smoothing and one backward Gauss Seidel sweep for post smoothing e MG W cycle n symmetric Gauss Seidel steps One iteration consists of one multigrid W cycle
5. and element residuals of a given element K take the form nk akl f div Dgradu a Vu bullZacic J ciaslilns DVulelliace ECOK where the weight ax is defined in the same way as the weight ap with the edge E replaced by the element K ALF has implemented two refinement strategies 1 the maximum strategy and 2 the equilibration strategy Both strategies assume that an error indicator nx has been computed for every element of a given partition 7 Both first mark a percentage EXCESS of the elements with largest ng for refinement This set of elements is labelled Mo the set of unmarked elements is labelled U In the maximum strategy an element K in U is marked for refinement if and only if nx gt THRESHOLD max 1K These elements form the set M In the equilibration strategy the set M is constructed by taking all elements of U with largest ng such that SO n gt THRESHOLD Y np KEM Keu The union of the sets Mo and M is the collection of the elements that are refined regularly In order to avoid hanging nodes additional 16 R VERFURTH elements are refined green or blue as described in the literature cf e g Chapter 4 of R Verftirth A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques Wiley Teubner 1996 or Chapters II 3 and II 4 of the Lecture Notes Numerische Behandlung von Differentialgleichungen II Finite Elemente The choice Mesh refinement allows you to choose among
6. settings of the text field Maximum number of nodes ALF requires about 50M N doubles and 60M N integers where MN is the number given in the text field Maximum number of nodes RUHR UNIVERSITAT BOCHUM FAKULTAT FUR MATHEMATIK D 44780 Bo CHUM GERMANY E mail address rv num1 rub de
7. the im plemented refinement and error estimation algorithms The choice Refinement parameters allows you to choose the pa rameters EXCESS and THRESHOLD described above and the pa rameter FIRSTADAPTIVE that determines the first level on which an adaptive refinement is performed The default settings are e EXCESS 0 1 e THRESHOLD 0 5 e FIRSTADAPTIVE 1 When choosing the option user settings you will be asked to enter these three parameters Figure 12 shows the mesh of Level 8 obtained for the example Convection diffusion interior amp boundary layer with the parameter 0 000001 and the default settings for EXCESS THRESHOLD and FIRSTADAPTIVE Figure 13 shows the mesh of Level 7 for the same example with the same parameters and THRESHOLD but with EXCESS 0 2 and FIRSTADAPTIVE 2 eoe ALF Graphics E xmin 1 0 Xmax 1 0 last level F grid Ell FIGURE 12 Mesh of Level 8 for a convection diffusion equation 8 NUMBER OF NODES AND LEVELS LF has two key parameters e ML the number of refinement levels and e MN the maximal number of nodes ALF USER GUIDE 17 eoe ALF Graphics last level F grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 FIGURE 13 Mesh of Level 7 for a convection diffusion equation These are set in the text fields Number of refinement levels and Maximal number of nodes respective
8. with a user specified number of symmetric Gauss Seidel sweeps for pre and post smoothing You will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps e MG V cycle 1 downwind Gauss Seidel step One iteration con sists of one multigrid V cycle with one downwind Gauss Seidel sweep for pre and post smoothing The unknowns are re ordered lexicographically with respect to the convection direc tion such that the stiffness matrix is close to lower diagonal form in the limit case of vanishing diffusion e MG V cycle n downwind Gauss Seidel steps One iteration consists of one multigrid V cycle with a user specified number of downwind Gauss Seidel sweeps for pre and post smoothing The unknowns are re ordered lexicographically with respect to the convection direction such that the stiffness matrix is close to lower diagonal form in the limit case of vanishing diffusion You will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps e MG W cycle 1 downwind Gauss Seidel step One iteration consists of one multigrid W cycle with one downwind Gauss Seidel sweep for pre and post smoothing The unknowns are re ordered lexicographically with respect to the convection di rection such that the stiffness matrix is close to lower diagonal form in the limit case of vanishing di
9. ALF USER GUIDE R VERFURTH CONTENTS 1 Introduction 1 2 User interface 2 3 Domain and initial grid 3 4 Differential equation 9 5 Discretization 12 6 Solver 12 7 Mesh refinement 14 8 Number of nodes and levels 16 9 Graphics 17 10 Error handling and memory requirement 19 1 INTRODUCTION ALF is an applet demonstrating Adaptive Linear Finite element methods for solving linear stationary convection diffusion reaction equations with Dirichlet and Neumann boundary conditions in two dimensional domains The finite element partitions consist of triangles and parallelograms both types may be mixed The finite element func tions are correspondingly linear or bilinear The partitions are obtained by successively refining an initial partition either uniformly or adap tively The adaptive refinement is based on residual a posteriori error estimators The discrete problems are obtained from a standard weak formulation of the differential equation using an SUPG stabilisation for the convective terms Multigrid or conjugate gradient algorithms are used for solving the discrete problems The partitions finite element solutions and error indicators can be visualised graphically The next section explains ALF s main window The following six sections show how to use the various choices and text fields that are displayed in the main window and that are used to choose domains initial grids differential equations solvers error indicators e
10. amples e g solving a Poisson equation with a sine series solution require additional input This will be provided via additional windows that show up after pressing the Compute Button of the main window These additional windows all have a top label explaining their purpose one or several labelled text fields for entering the relevant parameters and an OK button You have to press the latter one after having finished your input in order to continue the computation process Once the computation is completed you may eventually want to vi sualise the results by clicking the Draw Graph button Upon pressing the Draw Graph button a graphics window will show up see Fig 2 ALF USER GUIDE 3 It is split into three parts an upper part containing two choices and four text fields labelled Xmin Xmax Ymin and Ymax a lower part containing buttons labelled Draw and Quit and a large mid dle part for the graphics The values in the text fields determine the drawing area min Zmax X Ymin Ymax and are initially set to default values Note that abscissa and ordinate scale separately thus a circle will look like an ellipse if max min differs from Ymax Ymin Clicking the Draw button starts the graphics Clicking the Quit button termi nates the graphics closes the graphics window and leads back to the main window eoe ms ALF Graphics first level T grid WY xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 J 4 E 4
11. circle with radius 1 and centre at the origin The initial partition is obtained by replac ing the circle by an inscribed regular octahedron and joining its vertices with the origin When refining an element having an edge with its endpoints on the circle s boundary the midpoint of the edge will be projected onto the circle Figures 3 and 4 show for this example the initial partition and the partition resulting from 4 steps of uniform refinement eoe ALF Graphics first level F grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 _ 4 3 4 3 4 FIGURE 3 Initial partition of example Circle 8 triangles e Circle 4 squares 8 triangles The domain is the circle with radius 1 and centre at the origin The initial partition is shown in Figure 5 When refining an element having an edge with its endpoints on the circle s boundary the midpoint of the edge will be projected onto the circle e Segment 6 triangles The domain and initial partition are ob tained from those of the example Circle 8 triangles by remov ing the quarter segment in the quadrant x gt 0 y lt 0 and the ALF USER GUIDE ALF Graphics last level EP grid Xmax 1 0 W xmin 1 0 Ymin 1 0 Ymax 1 0 o oe ARXA SOSS ASAS EERE DS Ro TRO ATA KI N z XX CXXX AAA CHK KAAS SOK AR AER OSI LY d L b4 AY G 5 0 Ps 7 7 y 7 a PATAN
12. data are chosen such that the exact solution is given by 1 x 1 y Poisson equ with singular solution The equation is Au f with non homogeneous Dirichlet boundary conditions The force term f and the boundary data are chosen such that the exact solution is given by the singular solution for a 2_segment that takes in polar co ordinates the form r3 sin 3y When us ing adaptive refinement the mesh should be refined close to the origin Poisson equ with unit load The equation is Au 1 with homogeneous Dirichlet boundary conditions The exact solution is unknown Reaction diffusion equation interior layer The differential equation is Au bu f with non homogeneous Dirichlet boundary conditions The force term f the reaction term b and the boundary data are chosen such that the exact solution is given by tanh x y You will be asked to enter the parameter The solution exhibits an interior layer along the circle with radius z and cen tre at the origin When using adaptive refinement the mesh should be refined close to the boundary of this circle Discontinuous diffusion The differential equation is div Dgradu 1 with homogeneous Dirichlet boundary conditions The diffusion matrix D is piecewise constant in the region 4z 16y lt 1 its diagonal and off diagonal elements are and seu respectively outside this region it is the identity matrix You will be asked to enter the parame
13. ffusion e MG W cycle n downwind Gauss Seidel steps One iteration consists of one multigrid W cycle with a user specified number of downwind Gauss Seidel sweeps for pre and post smoothing The unknowns are re ordered lexicographically with respect to the convection direction such that the stiffness matrix is close to lower diagonal form in the limit case of vanishing diffusion You will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps e MG V cycle 1 squared Richardson step One iteration consists of one multigrid V cycle with one Richardson iteration for the squared system as pre and post smoothing step e MG V cycle n squared Richardson steps One iteration con sists of one multigrid V cycle with a user specified number of 14 R VERFURTH Richardson iterations for the squared system as pre and post smoothing step You will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps e MG W cycle 1 squared Richardson step One iteration consists of one multigrid W cycle with one Richardson iteration for the squared system as pre and post smoothing step e MG W cycle n squared Richardson steps One iteration con sists of one multigrid W cycle with a user specified number of Richardson iterations for the squared system as pre and post smoothing step You
14. he point lies in the interior of the domain label 0 or on a Dirichlet boundary label gt 0 or on a Neumann boundary label lt 0 The vertices of each sub domain must be ordered such that the interior of the sub domain lies on the left hand side when passing through its boundary according to this enumeration e Initial grid of previous example The next computation uses the same coarsest partition as in the previous run of ALF This option is particularly useful when using a user specified initial partition or when using the automatic tessellation facility 4 DIFFERENTIAL EQUATION The choice PDE provides the following sample differential equations e Poisson equ with sin series solution The equation is u f with non homogeneous Dirichlet boundary conditions The force term f and the boundary data are chosen such that the exact solution is a user specified sine series You will be asked for the number of terms the amplitudes and the frequencies 10 R VERFURTH e Poisson equ with cos series solution The equation is Au f with non homogeneous Dirichlet boundary conditions The force term f and the boundary data are chosen such that the exact solution is a user specified cosine series You will be asked for the number of terms the amplitudes and the frequencies Poisson equ with bubble solution The equation is Au f with non homogeneous Dirichlet boundary conditions The force term f and the boundary
15. her the prescribed number of refinement levels or the prescribed number of nodes is attained or when the allocated storage is exhausted It then prints one of the following messages in the output area of its main window Final level reached A standard exit Too many new nodes and too many new elements A standard exit e Too many new nodes A standard exit e Too many new elements A standard exit e Too many unknowns in stiffness matriz A less standard but harmless exit Too many non zero elements in stiffness matrix A less stan dard but harmless exit Iterative solver did diverge In this case you should try another solution algorithm When using a multigrid algorithm increas ing the number of smoothing steps and switching from a V to a W cycle might help No red elements found Indicates problems with the refinement strategy and should not appear Too small lu in stiffness matrix Should not appear Too large lv in stiffness matrix Should not appear 20 R VERFURTH Sometimes user input data may be erroneous ALF then prints an error message and stops the computation In this case you may try to compute the same example with a new input or a completely new example The following error messages may be encountered e Inconsistent neighbour relation Indicates an erroneous user input when using the options user defined initial grid or user defined boundary automatic tessellation e Inconsistent node data
16. id E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 4 o 4 12 gi 1 o Draw QUIT FIGURE 6 Initial partition of example Annulus 12 triangles e208 ALF Graphics last level F grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 Finest level 4 CX XX X K S oe KOO ee AA KA KK I A LA x CX i f gt TAA EXX KXLY E ARX REUNY DN AY NYY YY y CCCGGGE EI KAAAA AAA y SA A Sait XX XOX MAK OY 1 XOX SOS i KAA 3 5 Com JCT FIGURE 7 Partition resulting from 4 steps of uniform refinement applied to the grid of Fig 6 e Double ellipse 8 triangles The domain s boundary consists of two ellipses centred at the origin The top one has half axes 1 ALF USER GUIDE 7 and i the lower one has half axes 1 and L This domain can be obtained from the circle with radius 1 and centre at the origin by re scaling the abscissa by a factor 4 and F in the upper and lower half planes respectively The initial partition is obtained from the one of the example Circle 8 triangles by applying this re scaling When refining an element having an edge with its endpoints on the boundary the midpoint of the edge will be projected onto the corresponding ellipse Figures 8 and 9 show the initial grid and the mesh obtained after 4 steps of uniform refinement eoe _ ALF Graphics first level F grid E
17. ive refinement the mesh should be refined close to these regions Since the exponential layer is far stronger than the parabolic layer the refinement will first take place close to the region x 1 y gt 0 Convection diffusion 3 boundary layers The differential equa tion is cAu a Vu bu f in the square 1 1 x 1 1 with homogeneous Dirichlet boundary conditions The convection is given by a z y 0 1 27 the force term f takes the form 1 2x 1 y You will be asked to enter the parameter The exact solution of this problem is unknown But it is known that it exhibits an ex ponential boundary layer at the top boundary of the square and two parabolic boundary layers along the left and right bound aries When using adaptive refinement the mesh should be re fined close to these regions Since the exponential layer is far stronger than the parabolic layer the refinement will first take place close to the top boundary of the square 12 R VERFURTH 5 DISCRETIZATION The differential equation div Dgradu a Vu bu f is discretized by Su Dv a Yuu bw Q x f div Dgradu a Vu bu a Vu KET K fot Yo bx fa Vo if KET K The SUPG stabilisation parameters are given by 2 OK ix coun REM i lalla 2e lallzhx where is the smallest eigenvalue of the diffusion matrix D Non homogeneous Dirichlet boundary conditions are taken into account by homogenisation
18. k the partition of the next level is obtained either by a uniform refinement or by an adaptive refinement based on various a posteriori error indicators and refinement strategies This process is repeated until the prescribed number of refinement levels is reached or until the allocated storage is exhausted cf Sections 8 and 10 When using a uniform refinement every element is refined regularly i e the midpoints of edges are connected for triangles and the mid points of opposite edges are connected for quadrilaterals ALF has implemented two types of residual error indicators ALF USER GUIDE 15 e edge residuals and e edge and element residuals Given the differential equation div Dgradu a Vu bu f the edge residuals of an element K take the form Nk X 4ag ne DVunlellzece ECOK Here up is the solution of the discrete problem e denotes the geometric mean of the two eigenvalues of the diffusion matrix D evaluated at the element s barycentre ng is a unit vector orthogonal to the edge F g denotes the jump across the edge FE and the weight ag is given by Ap min hpe 2 B72 where Ap is the diameter of E and 8p is the maximum of 0 and b diva evaluated at the element s barycentre Edges on a Dirichlet boundary must be ignored For edges on a Neumann boundary the term ng DVunlz must be replaced by g n DVup where g is the given Neumann datum and n denotes the exterior unit normal to Q The edge
19. ly The refinement solution loop cf Section 7 is performed until the number ML of levels is reached or the number MN of nodes is at tained or the allocated storage which is a fixed factor of MN is exhausted cf Section 10 Choosing a too small number of ML or MN naturally leads to computations with only a small number of grids that do not allow for an adequate finite element solution Choosing a too large number of ML or most critical M N may cause troubles with the memory allocation of your Java virtual machine 9 GRAPHICS ALF s graphics window enables you to visualise the results of your computations In the top row of the window you ll see four text fields that deter mine the clipping area min Ymax X Ymin Ymax Usually you ll want to choose the values in these text fields such that the whole computa tional domain will be shown But selecting other values allows you to get a zoom of sub domains that might be of particular interest Note that abscissa and ordinate scale separately thus a circle will look like an ellipse if max Umin differs from Ymax Ymin Clicking the Draw button starts the graphics The left choice in the top row allows you to navigate through the various mesh levels e First level Go to the coarsest level e Next level Increase the level count by 1 e Previous level Decrease the level count by 1 18 R VERFURTH e Last level Go to the finest level The right choice in
20. nter for each node its z and y co ordinates and a label that specifies whether the point lies in the interior of the domain label 0 or on a Dirichlet bound ary label gt 0 or on a Neumann boundary label lt 0 Then you are asked for the number of elements Finally you have to enter for each element four integers giving the global num bers of the element s nodes When the element is a triangle the fourth number must be 1 The local enumeration of the element vertices must be such that the element interior lies on the left hand side when passing through the element boundary according to this enumeration User defined boundary automatic tessellation The domain is split into sub domains that are each specified by an ordered list ALF USER GUIDE 9 eoe ALF Graphics last level F grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 I MAAAAAMIAJ ARAL ARLES A AAAAAAAAAJ MAAAAAAAANAAAAAAAAAA AALAALIAAI E Draw QUIT FIGURE 11 Partition resulting from 4 steps of uniform refinement applied to the grid of Fig 10 of vertices Each sub domain is automatically partitioned us ing a chop off algorithm The sub domains and their partitions are automatically glued together First you will be asked for the number of sub domains For each sub domain you then have to specify the number of its vertices For each vertex you must enter its z and y co ordinates and a label that speci fies whether t
21. tc The graphics section explains the possibilities for visualising grids solutions and error indicators The last section finally discusses possible error Date September 2006 2 R VERFURTH messages and how to overcome these in particular when encountering problems with insufficient memory 2 USER INTERFACE Starting ALF you ll see a screen see Fig 1 which is split into three parts a left upper part labelled Input a right upper part labelled Output and a lower part containing buttons labelled Compute Draw Graph and Quit e098 ALF User Interface INPUT OUTPUT Domain and initial grid circle 8 triangles PDE Poisson equ with unit load Solver MG V cycle 1 2 symmetric Gauss Seidel step Solution parameters default number of refinement levels 4 maximal number of nodes 8000 EG 2 Compute w Graph C Quit FIGURE 1 ALF s main window In the input part you ll see several choices that allow to choose ex amples implemented methods etc and text fields for setting relevant parameters The text fields usually exhibit the default values of the cor responding parameters These choices and text fields will be explained in the following six sections Once you have made your choices and entered your favourite param eters you press the Compute button to start the computations The results will be depicted in the output part Note that some ex
22. ter K The exact solution of this problem is unknown but it is known that it exhibits an interior layer at ALF USER GUIDE 11 the discontinuity of the diffusion matrix When using adaptive refinement the mesh should be refined close to this region Convection diffusion interior layer The differential equation is cAut a Vu bu f with non homogeneous Dirichlet boundary conditions The con vection is given by a x y y x The force term f the reaction term b and the boundary data are chosen such that the exact solution is given by tanh e 2 x y You will be asked to enter the parameter The solution exhibits an interior layer along the circle with radius i and centre at the origin When using adaptive refinement the mesh should be refined close to the boundary of this circle Convection diffusion interior amp boundary layer The differen tial equation is cAut a Vut bu 0 in the square 1 1 x 1 1 with non homogeneous Dirich let boundary conditions The convection is given by a z y 2 1 7 the boundary condition is 0 at the left and top bound ary of the square and 100 at the right and bottom boundary You will be asked to enter the parameter The exact solution of this problem is unknown But it is known that it exhibits an exponential boundary layer at the boundary x 1 y gt 0 and a parabolic interior layer along the line connecting the points 1 1 and 1 0 When using adapt
23. the top row lets you choose the type of graphics e Grid Shows the grid of the actual level cf e g Figures 12 and 13 e Solution isolines Shows the isolines of the computed solution on the actual grid cf e g Figure 14 previous level F solution isoli ALF Graphics nes fi Xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 a er 4 g E T E S e 3 x F 4 T a p phu Ei S 3 KA E KL l Eog j d y a FIGURE 14 Isolines of a computed solution e Solution colourplot Shows a colourplot of the computed solu tion on the actual grid cf Figure 15 for the same solution as in Figure 14 eoe ALF Graphics next level solution colourplot I Xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 Draw QUIT FIGURE 15 Colourplot of the finite element function of Fig 14 ALF USER GUIDE 19 Estimated error Shows the estimated errors Each element is painted in a colour according to the relative size of the element s error indicator cf e g Figure 16 This option is only available in connection with an adaptive mesh refinement eoe ALF Graphics next level estimated error Xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 LELEL C Draw Quit FIGURE 16 Visualisation of error indicators 10 ERROR HANDLING AND MEMORY REQUIREMENT ALF stops its computations when eit
24. will be asked for the number of pre and post smoothing steps The number of pre smoothing steps may differ from the number of post smoothing steps e CG This is a classical conjugate gradient algorithm e CG with SSOR preconditioning This is a classical precondi tioned conjugate gradient algorithm with one symmetric suc cessive over relaxation sweep for preconditioning e BiCG Stab This is a stabilised bi conjugate gradient algorithm e BiCG Stab with SSOR preconditioning This is a stabilised and preconditioned bi conjugate gradient algorithm with one sym metric successive over relaxation sweep for preconditioning All solution algorithms terminate when either MAX IT iterations have been performed or when the initial residual measured in the Euclidean norm has been reduced by at least a factor TOL The parameter MAXIT is set to 10 for the multigrid algorithms with Gauss Seidel smoothing to 50 for the multigrid algorithms with squared Richardson smoothing and to 100 for the CG and BiCG algorithms The parameter TOL is set to 0 01 The choice Solution parameters gives you the opportunity to modify these parameters by choosing the option user settings the standard choice being default 7 MESH REFINEMENT ALF starts its computations on a partition which is labelled Level 1 and which is obtained by a uniform refinement of the coarsest mesh labelled Level 0 of the actual example Having computed the discrete solution on a mesh of Level
25. xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 Draw QUIT FIGURE 8 Initial partition of example Double ellipse 8 triangles eee ALF Graphics last level fi grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 C Draw Quit FIGURE 9 Partition resulting from 4 steps of uniform refinement applied to the grid of Fig 8 R VERFURTH e Double ellipse 4 squares 8 triangles The domain s boundary consists of two ellipses centred at the origin The top one has half axes 1 and i the lower one has half axes 1 and This domain can be obtained from the circle with radius 1 and centre at the origin by re scaling the abscissa by a factor z and in the upper and lower half planes respectively The initial parti tion is obtained from the one of the example Circle 4 squares 8 triangles by applying this re scaling When refining an ele ment having an edge with its endpoints on the boundary the midpoint of the edge will be projected onto the corresponding ellipse Figures 10 and 11 show the initial grid and the mesh obtained after 4 steps of uniform refinement eoe ALF Graphics first level F grid E xmin 1 0 Xmax 1 0 Ymin 1 0 Ymax 1 0 Draw Quit FIGURE 10 Initial partition of example Double ellipse 4 squares 8 triangles User defined initial grid You are first asked to enter the num ber of nodes Then you have to e
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