Moving mesh methods for solving parabolic partial differential equations

A new adaptive method is described for solving non-linear parabolic partial differential equations with moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is selected based upon the given initial data. The mesh movement at the boundary is governed by a second monitor function, which may or may not be the same as that used to drive the interior mesh movement. The method is described in detail and a selection of computational examples are presented using different monitor functions applied to the porous medium equation (PME) in one and two space dimensions.

[1]  Robert D. Russell,et al.  Moving Mesh Techniques Based upon Equidistribution, and Their Stability , 1992, SIAM J. Sci. Comput..

[2]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[3]  Peter K. Jimack,et al.  Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions , 2006 .

[4]  W. Strauss,et al.  Decay and scattering of solutions of a nonlinear Schrödinger equation , 1978 .

[5]  Robert D. Russell,et al.  Moving Mesh Methods for Problems with Blow-Up , 1996, SIAM J. Sci. Comput..

[6]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[7]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[8]  H. Eberl,et al.  Modeling and simulation of a bacterial biofilm that is controlled by pH and protonated lactic acids , 2008 .

[9]  J. F. Williams,et al.  MOVING MESH GENERATION USING THE PARABOLIC MONGE–AMPÈRE EQUATION∗ , 2008 .

[10]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[11]  P. Zegeling,et al.  Adaptive moving mesh computations for reaction--diffusion systems , 2004 .

[12]  Carol S. Woodward,et al.  Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers , 2020, ACM Trans. Math. Softw..

[13]  HuangWeizhang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[14]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[15]  J. F. Williams,et al.  Moving Mesh Generation Using the Parabolic Monge--Amp[e-grave]re Equation , 2009, SIAM J. Sci. Comput..

[16]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[17]  Chris Budd,et al.  An invariant moving mesh scheme for the nonlinear diffusion equation , 1998 .

[18]  Shrinivas Lankalapalli,et al.  An adaptive finite element method for magnetohydrodynamics , 1998, J. Comput. Phys..

[19]  J. F. Williams,et al.  Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions , 2006 .

[20]  Robert D. Russell,et al.  Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems , 1998, SIAM J. Sci. Comput..

[21]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[22]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[23]  K. Hensel.  Journal für die reine und angewandte Mathematik , 1892 .

[24]  Willem Hundsdorfer,et al.  An adaptive grid refinement strategy for the simulation of negative streamers , 2006, J. Comput. Phys..

[25]  C. Budd,et al.  The geometric integration of scale-invariant ordinary and partial differential equations , 2001 .

[26]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[27]  J. Mackenzie,et al.  A moving mesh method for the solution of the one-dimensional phase-field equations , 2002 .

[28]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[29]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[30]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[31]  Guojun Liao,et al.  A new approach to grid generation , 1992 .

[32]  Joel C. W. Rogers,et al.  NUMERICAL SOLUTION OF A DIFFUSION CONSUMPTION PROBLEM WITH A FREE BOUNDARY , 1975 .

[33]  J. Hansen,et al.  Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial differential equations , 1991 .

[34]  Banesh Hoffmann.  The Strange Story of the Quantum , 1959 .

[35]  Peter K. Jimack,et al.  An adaptive multigrid tool for elliptic and parabolic systems , 2005 .

[36]  Guojun Liao,et al.  Adaptive grid generation based onthe least-squares finite-element method , 2004 .

[37]  Blow-up in a Chemotaxis Model Using a Moving Mesh Method , 2009 .

[38]  Martin Berzins,et al.  A 3D UNSTRUCTURED MESH ADAPTATION ALGORITHM FOR TIME-DEPENDENT SHOCK-DOMINATED PROBLEMS , 1997 .

[39]  C C Pain,et al.  Anisotropic mesh adaptivity for multi-scale ocean modelling , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[40]  Georg Stadler,et al.  A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media , 2010, J. Comput. Phys..

[41]  Salim Meddahi,et al.  A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part II: a posteriori error analysis , 2004 .

[42]  A. Choudary,et al.  Partial Differential Equations An Introduction , 2010, 1004.2134.

[43]  F. Weissler.  Existence and non-existence of global solutions for a semilinear heat equation , 1981 .

[44]  M. Baines.  Moving finite elements , 1994 .

[45]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

[46]  Bharat K. Soni,et al.  Handbook of Grid Generation , 1998 .

[47]  J. A. White,et al.  On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems , 1979 .

[48]  M. Chari,et al.  Finite elements in electrical and magnetic field problems , 1980 .

[49]  J. Miller.  Numerical Analysis , 1966, Nature.

[50]  Leszek Demkowicz,et al.  Optimal error estimate of a projection based interpolation for the p-version approximation in three dimensions , 2005 .

[51]  Gian Luca Delzanno,et al.  An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization , 2008, J. Comput. Phys..

[52]  M. Aurada,et al.  Convergence of adaptive BEM for some mixed boundary value problem , 2012, Applied numerical mathematics : transactions of IMACS.

[53]  A. Wacher,et al.  String Gradient Weighted Moving Finite Elements in Multiple Dimensions with Applications in Two Dimensions , 2007, SIAM J. Sci. Comput..

[54]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[55]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[56]  Alireza Tabarraei,et al.  Adaptive computations on conforming quadtree meshes , 2005 .

[57]  Peter K. Jimack,et al.  A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries , 2005 .

[58]  Robert D. Russell,et al.  Adaptive mesh movement — the MMPDE approach and its applications , 2001 .

[59]  J. F. Williams,et al.  How to adaptively resolve evolutionary singularities in differential equations with symmetry , 2010 .

[60]  P. Brown.  A local convergence theory for combined inexact-Newton/finite-difference projection methods , 1987 .

[61]  J. Z. Zhu,et al.  The finite element method , 1977 .

[62]  A Best Approximation Property of the Moving Finite Element Method , 1996 .

[63]  Ruo Li,et al.  Moving Mesh Methods for Singular Problems on a Sphere Using Perturbed Harmonic Mappings , 2006, SIAM J. Sci. Comput..

[64]  Martin Berzins,et al.  SPRINT2D: adaptive software for PDEs , 1998, TOMS.

[65]  Martin Berzins,et al.  On spatial adaptivity and interpolation when using the method of lines , 1998 .

[66]  W. Strauss,et al.  Partial Differential Equations: An Introduction , 1992 .

[67]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[68]  G. Carey,et al.  Finite Elements: Fluid Mechanics Vol.,VI , 1986 .

[69]  Robert D. Russell,et al.  A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations , 2008, J. Comput. Phys..

[70]  Charbel Farhat,et al.  The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids , 2001 .

[71]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[72]  Weizhang Huang,et al.  Moving Mesh Methods Based on Moving Mesh Partial Differential Equations , 1994 .

[73]  Peter A. Markowich,et al.  Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit , 1999, Numerische Mathematik.

[74]  Weizhang Huang,et al.  A high dimensional moving mesh strategy , 1998 .

[75]  W. Hughes,et al.  Schaum's outline of theory and problems of fluid dynamics , 1991 .

[76]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[77]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[78]  Robert D. Russell,et al.  Adaptivity with moving grids , 2009, Acta Numerica.

[79]  J. Filo,et al.  The porous medium equation in a two-component domain , 2009 .

[80]  J. Reddy.  An introduction to the finite element method , 1989 .

[81]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[82]  Ashley Twigger,et al.  Blow-up in the Nonlinear Schrödinger Equation Using an Adaptive Mesh Method , 2008 .

[83]  WEIZHANG HUANG,et al.  A Moving Mesh Method Based on the Geometric Conservation Law , 2002, SIAM J. Sci. Comput..