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A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

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Abstract

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

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References

  1. Ahmadian, H., Friswell, M., Mottershead, J.: Minimization of the discretization error in mass and stiffness formulations by an inverse method. Int. J. Numer. Methods Eng. 41(2), 371–387 (1998)

    MATH  Google Scholar 

  2. Ainsworth, M., Wajid, H.A.: Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration. SIAM J. Numer. Anal. 48(1), 346–371 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Angel, J.B., Banks, J.W., Henshaw, W.D.: High-order upwind schemes for the wave equation on overlapping grids: Maxwell’s equations in second-order form. J. Comput. Phys. 352, 534–567 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Assêncio, D.C., Teran, J.M.: A second order virtual node algorithm for stokes flow problems with interfacial forces, discontinuous material properties and irregular domains. J. Comput. Phys. 250, 77–105 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Bedrossian, J., von Brecht, J.H., Zhu, S., Sifakis, E., Teran, J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains. J. Comput. Phys. 229(18), 6405–6426 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Bochkov, D., Gibou, F.: Solving poisson-type equations with robin boundary conditions on piecewise smooth interfaces. J. Comput. Phys. 376, 1156–1198 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: I. A stabilized lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199(41–44), 2680–2686 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Chen, L., Wei, H., Wen, M.: An interface-fitted mesh generator and virtual element methods for elliptic interface problems. J. Comput. Phys. 334, 327–348 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Colella, P., Graves, D.T., Keen, B.J., Modiano, D.: A cartesian grid embedded boundary method for hyperbolic conservation laws. J. Comput. Phys. 211(1), 347–366 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Crockett, R., Colella, P., Graves, D.: A cartesian grid embedded boundary method for solving the poisson and heat equations with discontinuous coefficients in three dimensions. J. Comput. Phys. 230(7), 2451–2469 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Dakin, G., Despres, B., Jaouen, S.: Inverse lax-wendroff boundary treatment for compressible lagrange-remap hydrodynamics on cartesian grids. J. Comput. Phys. 353, 228–257 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Dey, B., Idesman, A.: A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes. Part 2: numerical simulation and comparison with FEM. Arch. Appl. Mech. pp. 1–32 (2020)

  13. Fries, T., Omerović, S., Schöllhammer, D., Steidl, J.: Higher-order meshing of implicit geometries—Part I: Integration and interpolation in cut elements. Comput. Methods Appl. Mech. Eng. 313, 759–784 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Guddati, M.N., Yue, B.: Modified integration rules for reducing dispersion error in finite element method. Comput. Methods Appl. Mech. Eng. 193, 275–287 (2004)

    MATH  Google Scholar 

  15. Gyrya, V., Lipnikov, K.: M-adaptation method for acoustic wave equation on square meshes. J. Comput. Acoust. 20(1250022–1), 23 (2012)

    MathSciNet  MATH  Google Scholar 

  16. He, Z.C., Cheng, A.G., Zhang, G.Y., Zhong, Z.H., Liu, G.R.: Dispersion error reduction for acoustic problems using the edge-based smoothed finite element method (es-fem). Int. J. Numer. Methods Eng. 86(11), 1322–1338 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Hellrung, J.L., Wang, L., Sifakis, E., Teran, J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J. Comput. Phys. 231(4), 2015–2048 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Hoang, T., Verhoosel, C.V., Auricchio, F., van Brummelen, E.H., Reali, A.: Mixed isogeometric finite cell methods for the stokes problem. Comput. Methods Appl. Mech. Eng. 316, 400–423 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Hosseinverdi, S., Fasel, H.F.: An efficient, high-order method for solving poisson equation for immersed boundaries: combination of compact difference and multiscale multigrid methods. J. Comput. Phys. 374, 912–940 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Idesman, A.: Optimal reduction of numerical dispersion for wave propagation problems. Part 1: Application to 1-d isogeometric elements. Comput. Methods Appl. Mech. Eng. 317, 970–992 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Idesman, A.: The use of the local truncation error to improve arbitrary-order finite elements for the linear wave and heat equations. Comput. Methods Appl. Mech. Eng. 334, 268–312 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Idesman, A., Dey, B.: Optimal reduction of numerical dispersion for wave propagation problems. Part 2: Application to 2-d isogeometric elements. Comput. Methods Appl. Mech. Eng. 321, 235–268 (2017)

  23. Idesman, A., Dey, B.: The use of the local truncation error for the increase in accuracy of the linear finite elements for heat transfer problems. Comput. Methods Appl. Mech. Eng. 319, 52–82 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Idesman, A., Dey, B.: A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes. Comput. Methods Appl. Mech. Eng. 354, 568–592 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Idesman, A., Dey, B.: Accurate numerical solutions of 2-d elastodynamics problems using compact high-order stencils. Comput. Struct. 229, 1–18 (2020)

    Google Scholar 

  26. Idesman, A., Dey, B.: Accurate numerical solutions of 2-d elastodynamics problems using compact high-order stencils. Comput. Struct. 229, 106160 (2020)

    Google Scholar 

  27. Idesman, A., Dey, B.: Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equations. Comput. Methods Appl. Mech. Eng. 360, 112699 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Idesman, A., Dey, B.: New 25-point stencils with optimal accuracy for 2-d heat transfer problems. Comparison with the quadratic isogeometric elements. J. Comput. Phys. 418, 109640 (2020)

    MathSciNet  Google Scholar 

  29. Idesman, A., Dey, B.: A new numerical approach to the solution of the 2-D Helmholtz equation with optimal accuracy on irregular domains and Cartesian meshes. Comput. Mech. 65, 1189–1204 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Idesman, A., Dey, B.: The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes. Comput. Methods Appl. Mech. Eng. 365, 112985 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Idesman, A., Pham, D.: Accurate finite element modeling of acoustic waves. Comput. Phys. Commun. 185, 2034–2045 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Idesman, A., Pham, D.: Finite element modeling of linear elastodynamics problems with explicit time-integration methods and linear elements with the reduced dispersion error. Comput. Methods Appl. Mech. Eng. 271, 86–108 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Idesman, A., Schmidt, M., Foley, J.R.: Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error. Comput. Mech. 47, 555–572 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Johansen, H., Colella, P.: A cartesian grid embedded boundary method for poisson’s equation on irregular domains. J. Comput. Phys. 147(1), 60–85 (1998)

    MathSciNet  MATH  Google Scholar 

  35. Jomaa, Z., Macaskill, C.: The embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions. J. Comput. Phys. 202(2), 488–506 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Jomaa, Z., Macaskill, C.: The shortley-weller embedded finite-difference method for the 3d poisson equation with mixed boundary conditions. J. Comput. Phys. 229(10), 3675–3690 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Kreiss, H.O., Petersson, N.A.: A second order accurate embedded boundary method for the wave equation with dirichlet data. SIAM J. Sci. Comput. 27(4), 1141–1167 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Kreiss, H.O., Petersson, N.A., Ystrom, J.: Difference approximations of the neumann problem for the second order wave equation. SIAM J. Numer. Anal. 42(3), 1292–1323 (2004)

    MathSciNet  MATH  Google Scholar 

  39. Kreisst, H.O., Petersson, N.A.: An embedded boundary method for the wave equation with discontinuous coefficients. SIAM J. Sci. Comput. 28(6), 2054–2074 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Krenk, S.: Dispersion-corrected explicit integration of the wave equation. Comput. Methods Appl. Mech. Eng. 191, 975–987 (2001)

    MathSciNet  MATH  Google Scholar 

  41. Main, A., Scovazzi, G.: The shifted boundary method for embedded domain computations. Part I: Poisson and stokes problems. J. Comput. Phys. 372, 972–995 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Marfurt, K.J.: Accuracy of finite difference and finite element modeling of the scalar and elastic wave equation. Geophysics 49, 533–549 (1984)

    Google Scholar 

  43. Mattsson, K., Almquist, M.: A high-order accurate embedded boundary method for first order hyperbolic equations. J. Comput. Phys. 334, 255–279 (2017)

    MathSciNet  MATH  Google Scholar 

  44. May, S., Berger, M.: An explicit implicit scheme for cut cells in embedded boundary meshes. J. Sci. Comput. 71(3), 919–943 (2017)

    MathSciNet  MATH  Google Scholar 

  45. McCorquodale, P., Colella, P., Johansen, H.: A cartesian grid embedded boundary method for the heat equation on irregular domains. J. Comput. Phys. 173(2), 620–635 (2001)

    MathSciNet  MATH  Google Scholar 

  46. Mullen, R., Belytschko, T.: Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation. Int. J. Numer. Methods Eng. 18, 11–29 (1982)

    MathSciNet  MATH  Google Scholar 

  47. Puzyrev, V., Deng, Q., Calo, V.: Dispersion-optimized quadrature rules for isogeometric analysis: Modified inner products, their dispersion properties, and optimally blended schemes. Comput. Methods Appl. Mech. Eng. 320, 421–443 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Rank, E., Kollmannsberger, S., Sorger, C., Duster, A.: Shell finite cell method: a high order fictitious domain approach for thin-walled structures. Comput. Methods Appl. Mech. Eng. 200(45–46), 3200–3209 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., Duster, A.: Geometric modeling, isogeometric analysis and the finite cell method. Comput. Methods Appl. Mech. Eng. 249–252, 104–115 (2012)

    MATH  Google Scholar 

  50. Schwartz, P., Barad, M., Colella, P., Ligocki, T.: A cartesian grid embedded boundary method for the heat equation and poisson’s equation in three dimensions. J. Comput. Phys. 211(2), 531–550 (2006)

    MathSciNet  MATH  Google Scholar 

  51. Seriani, G., Oliveira, S.P.: Optimal blended spectral-element operators for acoustic wave modeling. Geophysics 72(5), 95–106 (2007)

    Google Scholar 

  52. Singh, K., Williams, J.: A parallel fictitious domain multigrid preconditioner for the solution of poisson’s equation in complex geometries. Comput. Methods Appl. Mech. Eng. 194(45–47), 4845–4860 (2005)

    MATH  Google Scholar 

  53. Song, T., Main, A., Scovazzi, G., Ricchiuto, M.: The shifted boundary method for hyperbolic systems: embedded domain computations of linear waves and shallow water flows. J. Comput. Phys. 369, 45–79 (2018)

    MathSciNet  MATH  Google Scholar 

  54. Uddin, H., Kramer, R., Pantano, C.: A cartesian-based embedded geometry technique with adaptive high-order finite differences for compressible flow around complex geometries. J. Comput. Phys. 262, 379–407 (2014)

    MathSciNet  MATH  Google Scholar 

  55. Vos, P., van Loon, R., Sherwin, S.: A comparison of fictitious domain methods appropriate for spectral/hp element discretisations. Comput. Methods Appl. Mech. Eng. 197(25–28), 2275–2289 (2008)

    MathSciNet  MATH  Google Scholar 

  56. Wang, D., Liu, W., Zhang, H.: Novel higher order mass matrices for isogeometric structural vibration analysis. Comput. Methods Appl. Mech. Eng. 260, 92–108 (2013)

    MathSciNet  MATH  Google Scholar 

  57. Wang, D., Liu, W., Zhang, H.: Superconvergent isogeometric free vibration analysis of euler-bernoulli beams and kirchhoff plates with new higher order mass matrices. Comput. Methods Appl. Mech. Eng. 286, 230–267 (2015)

    MathSciNet  MATH  Google Scholar 

  58. Wang, D., Liang, Q., Wu, J.: A quadrature-based superconvergent isogeometric frequency analysis with macro-integration cells and quadratic splines. Comput. Methods Appl. Mech. Eng. 320, 712–744 (2017)

    MathSciNet  MATH  Google Scholar 

  59. Yue, B., Guddati, M.N.: Dispersion-reducing finite elements for transient acoustics. J. Acoust. Soc. Am. 118(4), 2132–2141 (2005)

    Google Scholar 

  60. Zhao, S., Wei, G.W.: Matched interface and boundary (mib) for the implementation of boundary conditions in high-order central finite differences. Int. J. Numer. Methods Eng. 77(12), 1690–1730 (2009)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research has been supported in part by the Air Force Office of Scientific Research (contract FA9550-16-1-0177), by NSF (Grant CMMI-1935452) and by Texas Tech University.

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  1. Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, 79409-1021, USA

    A. Idesman

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Appendix A: The coefficients \(b_p\) used in Eqs. (38), (56), (63) and (69)

Appendix A: The coefficients \(b_p\) used in Eqs. (38), (56), (63) and (69)

The first five coefficients \(b_i\) (\(i=1,2,\ldots ,5\)) used in Eqs. (38), (56), (63) and (69) are presented below. All coefficients \(b_i\) used in these formulas are given in the attached file ’b-coeff.pdf’

Equation (38):

$$\begin{aligned} b_1= & {} (k_1 + k_2 + k_3 + k_4 + k_5 + k_6 + k_7 + k_8 + k_9), \nonumber \\ b_2= & {} (-d_1 k_1 + d_3 k_3 - d_4 k_4 + d_5 k_6 - d_6 k_7 + d_8 k_9), \nonumber \\ b_3= & {} b_y (-d_1 k_1 - d_2 k_2 - d_3 k_3 + d_6 k_7 + d_7 k_8 + d_8 k_9), \nonumber \\ b_4= & {} (d_1^2 k_1 + d_3^2 k_3 + d_4^2 k_4 + d_5^2 k_6 + d_6^2 k_7 + d_8^2 k_9\nonumber \\&+ 2 (m_1 + m_2 + m_3 + m_4 + m_5 + m_6 + m_7 + m_8 + m_9)), \nonumber \\ b_5= & {} 2 b_y (d_1^2 k_1 - d_3^2 k_3 - d_6^2 k_7 + d_8^2 k_9), \nonumber \\&\ldots \qquad \end{aligned}$$
(A.1)

Equation (56):

$$\begin{aligned} b_{1} \,= & {} \, (k_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6}+k_{8}+k_{9}), \nonumber \\ b_{2} \,= & {} \, - (k_{1}-k_{3}+k_{4}-k_{6}-k_{9}+l_1 n_{11}+l_2 n_{12}+l_3 n_{13}), \nonumber \\ b_{3} \,= & {} \, - (k_{1}+k_{2}+k_{3}-k_{8}-k_{9}+l_1 n_{21}+l_2 n_{22}+l_3 n_{23}), \nonumber \\ b_{4} \,= & {} \, \frac{1}{2} (k_{1}+k_{3}+k_{4}+k_{6}+k_{9}+2 m_{1}+2 m_{2}+2 m_{3}+2 m_{4}+2 m_{5}+2 m_{6}+2 m_{8}+2 m_{9} \nonumber \\&-2 \alpha _{1} l_1 n_{11}-2 \alpha _{2} l_2 n_{12}-2 \alpha _{3} l_3 n_{13}),\nonumber \\ b_{5} \,= & {} \, (k_{1}-k_{3}+k_{9}-\beta _1 l_1 n_{11}-\beta _2 l_2 n_{12}-\beta _3 l_3 n_{13}-\alpha _{1} l_1 n_{21}-\alpha _{2} l_2 n_{22}-\alpha _{3} l_3 n_{23}),\nonumber \\&\ldots \end{aligned}$$
(A.2)

Equation (63):

$$\begin{aligned} b_1 \,= & {} \, (k_1 + k_2 + k_3 + k_4 + k_5 + k_6 + k_7 + k_8 + k_9) , \nonumber \\ b_2 \,= & {} \, (-d_1 k_1 + d_3 k_3 - d_4 k_4 + d_5 k_6 - d_6 k_7 + d_8 k_9), \nonumber \\ b_3 \,= & {} \, b_y (-d_1 k_1 - d_2 k_2 - d_3 k_3 + d_6 k_7 + d_7 k_8 + d_8 k_9), \nonumber \\ b_4 \,= & {} \, 2 b_y (d_1^2 k_1 - d_3^2 k_3 - d_6^2 k_7 + d_8^2 k_9), \nonumber \\ b_5 \,= & {} \, (d_1^2 (-1 + b_y^2) k_1 - d_3^2 k_3 - d_4^2 k_4 - d_5^2 k_6 - d_6^2 k_7 - d_8^2 k_9 \nonumber \\&+ b_y^2 (d_2^2 k_2 + d_3^2 k_3 + d_6^2 k_7 + d_7^2 k_8 + d_8^2 k_9)),\nonumber \\&\ldots \end{aligned}$$
(A.3)

Equation (69):

$$\begin{aligned} \ b_{1} \,= & {} \, k_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6}+k_{8}+k_{9}, \nonumber \\ b_{2} \,= & {} \, -k_{1}+k_{3}-k_{4}+k_{6}+k_{9}+{l_1} n_{11}+{l_2} n_{12}+{l_3} n_{13}+{l_4} n_{14}, \nonumber \\ b_{3} \,= & {} \, -b_y (k_{1}+k_{2}+k_{3}-k_{8}-k_{9})+{l_1} n_{21}+{l_2} n_{22}+{l_3} n_{23}+{l_4} n_{24}, \nonumber \\ b_{4} \,= & {} \, b_y (k_{1}-k_{3}+k_{9}+{\beta _1} {l_1} n_{11}+{l_1} n_{11}+{\beta _2} {l_2} n_{12}+{l_2} n_{12}+{\beta _3} {l_3} n_{13}+{l_3} n_{13}+{\beta _4} {l_4} n_{14} \nonumber \\&+{l_4} n_{14})+({\alpha _1}-1) {l_1} n_{21}+{\alpha _2} {l_2} n_{22}-{l_2} n_{22}+{\alpha _3} {l_3} n_{23}-{l_3} n_{23}+{\alpha _4} {l_4} n_{24}-{l_4} n_{24}, \nonumber \nonumber \\ b_{5} \,= & {} \, \frac{1}{2} ((k_{2}+k_{3}+k_{8}+k_{9}) b_y^2+2 (({\beta _1}+1) {l_1} n_{21}+({\beta _2}+1) {l_2} n_{22}+{\beta _3} {l_3} n_{23}+{l_3} n_{23}+{\beta _4} {l_4} n_{24} \nonumber \\&+{l_4} n_{24}) b_y+(b_y^2-1) k_{1}-k_{3}-k_{4}-k_{6}-k_{9}-2 {\alpha _1} {l_1} n_{11}+2 {l_1} n_{11}-2 {\alpha _2} {l_2} n_{12} \nonumber \\&+2 {l_2} n_{12}-2 {\alpha _3} {l_3} n_{13}+2 {l_3} n_{13}-2 {\alpha _4} {l_4} n_{14}+2 {l_4} n_{14}), \nonumber \\&\ldots \end{aligned}$$
(A.4)

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Idesman, A. A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases. Arch Appl Mech 90, 2621–2648 (2020). https://doi.org/10.1007/s00419-020-01744-w

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