A mesh-free partition of unity method for diffusion equations on complex domains

A mesh-free partition of unity method for diffusion equations on complex domains
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   IMA Journal of Numerical Analysis  (2005) Page 1 of 27doi: 10.1093/imanum/dri017 A Meshfree Partition of Unity Method for Diffusion Equations onComplex Domains M. EIGEL ∗ , E. GEORGE†, M. KIRKILIONIS‡  Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL, UK. [Received on xxx] We present a numerical method for solving partial differential equations on domains with distinctivecomplicated geometrical properties. These will be called  complex domains . Such domains occur in manyreal world applications, for example in geology or engineering. We are however in particular interested inapplications stemming from the life sciences, especially cell biology. In this area such complex domainsas retrieved from microscopy images at different scales are the norm and not the exception. Therefore,geometry is expected to directly influence the physiological function of different systems, for examplesignalling pathways. New numerical methods able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restric-tions in the past. The approximation approach presented here for such problems is based on a promisingmesh-free Galerkin method, the  partition of unity method   (PUM). We introduce the main approximationfeatures, and then focus on the construction of appropriate covers as the basis of discretisations. Asa main result we present an extended version of cover construction ensuring fast convergence rates inthe solution process. Parametric patches are introduced as a possible way of approximating complicatedboundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergencebehaviour of the PUM is demonstrated in several numerical examples. Keywords : complex domains, meshfree partition of unity method, generalised finite elements, cover gen-eration 1. Introduction Simulation of technical and natural phenomena has become a primary tool of scientific understanding byeither pre- or re-constructing the interacting mechanisms of a given problem. The need to understand agrowing number of increasingly complicated problems has generated different sophisticated numericalmethods to approximate solutions of the associated equations. Many such problems typically involvingtransport and reactions, can in general be adequately described by partial differential equations (PDE),although there are many exceptions where stochastic effects and discrete entities do not allow exclu-sive concentration on spatial continuum approximations. Common and at the same time traditionalapproaches to numerically solving PDE are finite differences (FD) and finite element methods (FEM).Both depend on a fixed structure of discretisation nodes and a defined connectivity to neighbour nodes,given by a mesh.Research to generalise these methods and alleviate or entirely remove the dependency on a mesh has ∗  †  ‡  IMA Journal of Numerical Analysis c  Institute of Mathematics and its Applications 2005; all rights reserved.  2 of 27  M. EIGEL, E. GEORGE, M. KIRKILIONIS gained momentum since the 1990s with the development of moving least squares (MLS) methods andthe partition of unity method (PUM). Comprehensive overviews of mesh-free methods (MM) can befound in [9, 5, 17, 16]. The most important goals for MM is an improved flexibility in the constructionof local approximations and better handling of complicated geometries and refinements. In Sec. 2we describe the partition of unity mesh-free method (PUM) which is used throughout the paper. Itis implemented in a computational code, the  Generic Discretisation Framework   ( GDF  ), written bythe authors. A key aspect of the PUM, the construction of a cover of the domain, is explained indetail in Sec. 3. As a major extension to existing algorithms and central to this paper we discuss covercreation strategies in settings with complex geometries and holes. This is followed in Sec. 4 with anoverview of the actual discretisation process for a generic linear scalar PDE. Special attention is paidto the accurate treatment of boundary patches where numerical integration can degrade the convergenceorder. Finally, numerical examples demonstrate the capabilities of the method in Sec. 5. We confirmthe anticipated convergence features, which in particular are shown for a geometrically complex setupincluding several holes in the domain. The outlook section, Sec. 6, summarises the main ideas of themanuscript and discusses the necessary future extensions for simulating systems of coupled nonlineartransport equations on complex and, in particular, nested domains. 2. The Partition of Unity Method 2.1  Introduction The numerical method used in this paper is based on the framework of the partition of unity method(PUM), initially introduced by Melenk, Babuˇska [2, 18]. With regard to the construction of local func- tion spaces, the class of generalised finite element methods (GFEM) is closely related and can be seen asa subclass of the PUM. Application of PUM and GFEM to real world problems were described in a se-ries of papers by Griebel and Schweitzer [11, 15, 10, 13, 14, 12], the monograph [20] and in Strouboulis [25, 26, 23, 24]. The PUM exhibits some unique features which renders it especially attractive for someproblems classical FEM cannot deal with efficiently: •  Approximation spaces of arbitrary smoothness can be created. Moreover, special functions suitedfor approximating the sought solution are incorporated and adapted easily (cf. [18] and also [4]). •  No mesh has to be created for the discretisation. Meshfree methods usually are based on a set of nodes {  x i }  N i = 1  for which corresponding patches { ω  i }  N i = 1  are defined, usually with the property thatthe domain  Ω   is covered by the union of the patches. The nodes can be redistributed or the nodedensity changed without geometrical constraints as is the case with triangulations (cf. [8, 13]). •  Adaptivity by  hp -refinement can be carried out without any complications induced by the domainor function spaces (see [8, 11]).As usual we assume an appropriate Hilbert space H   ( Ω  )  and the variational form of the PDE writtenwith the help of a continuous bilinear form  a  and a linear form  l , i.e. a  : H    × H    → R  and  l  ∈ H   ′ , together with appropriate boundary condition. The problem we are concerned with can then be statedas follows:Find  u ∈ H    s.t.  a ( u , v ) =  l ( v )  ∀ v ∈ H   .  (2.1)The basic method of discretisation in the PUM framework is then given by the following steps:  A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains  3 of 27 •  Given  N   pointsinadomain Ω   onwhichalinearscalarPDEisdefined, opensetscalled  patches areassociated with each point to form a cover  Ω   N   of the domain. ( Ω   N   : = { ω  i }  N i = 1 , with  Ω   ⊂  i ω  i ). •  A partition of unity of consistency order 0 and arbitrary smoothness)  { ϕ  i }  N i = 1  subordinate to thecover is constructed. •  The local function space on patch  ω  i , 1  i   N  , is given by span  ψ  k i   p i k  = 1 , with  { ψ  k i  }  p i k  = 1  beinga set of approximating functions chosen for each patch. The global approximation space, alsocalled the trial or the  PUM space , is defined by V   PU  : = span { ϕ  i (  x ) ψ  k i  (  x ) } i , k  . Replacing H    by thefinite dimensional subspace  V   PU , a global approximation  u PU   to the unknown solution  u  of thePDE is defined as a (weighted) sum of local approximation functions on the patches: u PU (  x ) =  N  ∑ i = 1 ϕ  i (  x ) ψ  i (  x ) =  N  ∑ i = 1 ϕ  i (  x )   p i ∑ k  = 1 ξ k i  ψ  k i  (  x )  . In case of Neumann problems, as investigated in this article, the PUM approximation space isconforming, i.e.  V   PU   ⊂ H   . Dirichlet problems were treated in, for example, [10]. •  The unknown coefficients  ξ k i  are determined by substituting the above approximation into thePDE and using the method of weighted residuals to derive an algebraic system of equations  A ξ  =  b .  (2.2)Dirichlet boundary conditions are not discussed in this document as they are rarely encounteredin problems arising in cell biology. They are more complicated to deal with than Neumann boundaryconditions in mesh-free methods as they have to be imposed (weakly) on boundary patches. This isdue to the fact that the trial functions are not fulfilling the so called  Kronecker delta property . Differentapproaches have been suggested to handle essential boundary conditions. The  Nitsche method  , whichhas been used in the context of mortar finite elements for many years, and which also is used in somediscontinuous Galerkin methods, is one of the preferred approaches, see [10] for details and [3, 6] for discussions of other methods.2.2  Partition of Unity Method  Key to the PUM is the notion of a  (  M  , C  ∞ , C  ∇ ) -partition of unity. According to Babuˇska and Melenk [18] we define: Definition 2.1 (Partition of Unity)  Let Ω   ⊂ R d  be an open set, and let Ω   N   : = { ω  i }  N i = 1  be an open coverof   Ω   satisfying a point-wise overlap condition: ∃  M   ∈ N :  ∀  x  ∈ Ω   card { i  |  x  ∈ Ω  i }   M  . Moreover let  { ϕ  i }  be a Lipschitz partition of unity subordinate to the cover  Ω   N   satisfyingsupp ( ϕ  i ) ⊂ Ω   N   ∀ i ,  ∑ i ϕ  i  ≡ 1 on  Ω  ,  ϕ  i   L 2 ( R d  )  C  ∞ ,   ∇ ϕ  i   L ∞ ( R d  )   C  ∇ diam ( ω  i ) ,  4 of 27  M. EIGEL, E. GEORGE, M. KIRKILIONIS with two positive constants C  ∞  and C  ∇ . Then  { ϕ  i }  N i = 1  is called a  (  M  , C  ∞ , C  ∇ ) -partition of unity  subordi-nate to the cover  { ω  i } . The PU is said to be of degree  m ∈ N 0  if   ϕ  i  ∈ C  m ( R d  )  for all  i .There are several ways for constructing a partition of unity satisfying this definition. We use thewell known  Shepard functions  which are described later in this paper. In order to provide good (global)approximation results, the numerical method has to fulfil two main criteria: Local Approximation:  On each patch  ω  i  a local approximation space V   i  =  span { ψ  k i  }  p i k  = 1  of arbitrarydegree  p i  is defined. Functions in V   i  have to be able to approximate the solution in ω  i  accurately.Polynomial bases are most common but may not provide the best possible approximation forspecific problems, see [2, 18, 3] for further examples. Global Continuity:  The global approximation space  V   PU  is created by blending together the set of local spaces  { V   i }  N i = 1  by a partition of unity  { ϕ  i }  N i = 1  on Ω   while keeping the local approximationproperties. Definition 2.2 (PUM space)  Let  { ω  i }  N i = 1  be an open cover of   Ω   ⊂ R d  , let  { ϕ  i }  N i = 1  be a  (  M  , C  ∞ , C  ∇ ) -partition of unity subordinate to  { ω  i }  and let V   i  ⊂  H  1 ( Ω   ∩ ω  i )  be given. Then the space V   PU  : =  N  ∑ i = 1 ϕ  i V   i  =   N  ∑ i = 1 ϕ  i v  |  v ∈ V   i   (2.4) =  span { ϕ  i ψ  k i  |  ψ  k i  ∈ V   i , i  =  1 ...  N  , k   =  1 ...  p i } is called a  PUM space  of degree  m ∈ N if  V   PU  ⊂ C  m ( Ω  ) .The PUM space is used as approximation space for the discretisation of the problem equation. Ageneral approximation result is provided by the following theorem.T HEOREM  2.1 (A PPROXIMATION  P ROPERTY  [18]) Let  Ω   ⊂  R d  be given,  { ω  i } , { ϕ  i }  and  { V   i }  asin Definitions 2.2 and 2.1. Let  u  ∈ H    be the function to be approximated and assume that the localapproximation spaces V   i  possesses the following approximation properties: On each patch  Ω   ∩ ω  i  thefunction  u  can be approximated by a function  v i  ∈ V   i  such that  u − v i   L 2 ( Ω  ∩ ω  i )  ε  1 , i  and   ∇ ( u − v i )   L 2 ( Ω  ∩ ω  i )  ε  2 , i hold. Then the function  u PU   : = ∑ i ϕ  i v i  ∈ V   PU  ⊂ H    satisfies  u − u PU    L 2 ( Ω  )  √   MC  ∞  ∑ i ε  21 , i  1 / 2 ,  (2.5a)  ∇ ( u − u PU  )   L 2 ( Ω  )  √  2  M   ∑ i   C  ∇ diam ( ω  i )  2 ε  21 , i  + C  2 ∞ ε  22 , i  1 / 2 .  (2.5b)For instance, if   u  ∈  H  2 ( Ω  ) , assuming a patch-wise error bound depending on the patch size  h i  andlinear polynomial local spaces such that ε  1 , i ( h i ,  p i )  C  i h 2 i  p − 1 i   u   H  2 ( Ω  ∩ ω  i ) ,  A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains  5 of 27then Theorem 2.1 yields  u − u PU    L 2 ( Ω  )   MC  ∞ max i { C  i h 2 i  p − 1 i  } u   H  2 ( Ω  ∩ ω  i ) . This shows that the method can be used as a  h -,  p - or  hp -version by adjusting density of nodes orthe consistency order of the local spaces which makes the method very versatile. While this a prioriapproximation result is shown for polynomial spaces, it is obvious from the construction of the localspaces that the approximation can be improved if a base of the solution space of the problem is known,see [18]. Refinement strategies ( h -refinement by decreasing patch sizes or  p -refinement by increasinglocal approximation quality) can be carried out independently on each patch. Moreover, the burden tofulfil strict structuring criteria (e.g. avoiding hanging nodes and overlapping meshes) when dealing withrefinement and remeshing of triangulations is entirely omitted. The partition of unity ensures smooth-ness of the global function space while local features of the employed function spaces are preserved.When using an arbitrary cover of patches it can happen that the union of local bases is not linearlyindependent on some patches. While this might not be a major problem as long as only a few functionspaces exhibit linear dependence, the solution process may become more difficult. This complication of obtaining singular operator matrices was reported on in [24], where also an iterative solution algorithmwas suggested based on perturbing the algebraic system. A recent investigation of this topic can befound in [27].By the cover construction algorithm described in section 3 we completely abolish this difficulty byensuring linearly independent function spaces. In [11] Schweitzer defined the so called  flat top property which is a sufficient (but not necessary) condition for this. First, for a cover  Ω   N   : =  { ω  i }  N i = 1  we definethe neighbour index sets n (  x )  : = { i  |  x  ∈ ω  i } ∀  x  ∈ Ω  , n ( i )  : = {  j  |  ω  i ∩ ω   j   =  / 0 }  i  =  1 ,...,  N  . Definition 2.3 (Flat top property)  Let  Φ   : =  { ϕ  i }  be a partition of unity subordinate to the cover { ω  i }  N i = 1  of domain  Ω   as defined in 2.2,  µ   the Lebesgue measure in R d  . Then  Φ   is said to have the  flat top property , if  µ  ( ω  i )   ˜ C  µ  (  ˜ ω  i ) ,  i  =  1 ...  N   (2.6)with a constant ˜ C   ∈ R + independent of the patch and the sub-patch ˜ ω  i  ⊂ ω  i  defined as˜ ω  i  : = {  x  ∈ ω  i  |  n (  x ) = { i }} .  (2.7)R EMARK  2.1 In a partition of unity exhibiting this property all patches  ω  i  have a subset ˜ ω  i  larger thana null-set in the Lebesgue sense with no other patches overlapping.2.3  Partition of unity construction We now construct a straightforward 0th order PU due to  Shepard   [21]. First,  weight functions w i  : R d  → [ 0 , 1 ]  are defined on each patch. These are used to localise functions by requiring supp ( w i ) ⊂ ω  i .
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