Abstract
It is shown how the polynomial splines defined on the curved triangulations introduced in our recent paper Larry L. Schumaker, A. Yu. (Comput. Aided Geom. Design. 92:102050, 2022) can be used to solve elliptic PDEs defined on curved planar domains. The approach is similar to that used in Larry L. Schumaker (J. Sci. Comp. 80:1369–1394, 2019), but does not require immersing the domain of interest in a larger computational domain. The methods are easy to implement using Bernstein–Bézier representations of the splines, and the solutions are computed by solving certain penalized least-squares problems. A number of numerical examples are included to illustrate the performance of the methods.
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References
Apprich, C., Höllig, K., Hörner, J., Reif, U.: Collocation with WEB-splines. Adv. Comput. Math. 42, 823–842 (2016)
Barrett, J.W., Eilliot, C.M.: Finite element approximation of the Dirichet problem using the boundary penalty method. Numer. Math. 49, 343–366 (1986)
Bochev, P., Gunzburger, M.: Least-squares finite element methods, In: Int. Congress Mathematicians. Vol. III, Eur. Math. Soc. Zürich, pp. 1137–1162 (2006)
Bochev, P.B., Gunzburger, M.D.: Least-squares Finite Element Methods. In: Applied Mathematical Sciences, vol. 166. Springer, New York (2009)
Clark, B.W., Anderson, D.C.: The penalty boundary method. Finite Elem. Anal. Des. 39, 387–401 (2003)
Davydov, O., Kostin, G., Saeed, A.: Polynomial finite element method for domains enclosed by piecewise conics. Comput. Aided Geom. Design 45, 48–72 (2016)
Davydov, O., Saeed, A.: \(C^1\) quintic splines on domains encosed by piecewise conics and numerical solution of fully nonlinear elliptic equations. J. Appl. Num. Anal. 116, 172–183 (2017)
Eason, E.D.: A review of least-squares methods for solving partial differential equations. Int. J. Numer. Methods Engrg. 10, 1021–1046 (1976)
Eason, E.D., Mote, C.D.: Solution of non-linear boundary value problems by discrete least squares. Int. J. Numer. Methods Engrg. 11, 641–652 (1977)
Höllig, K.: Finite Element Methods with B-splines. SIAM, Philadelphia (2003)
Höllig, K., Reif, U., Wipper, J.: Weighted extended B-spline approximation of Dirichlet problems. SIAM J. Numer. Anal. 39, 442–462 (2001)
Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)
Schumaker, L.L.: Spline Functions: Computational Methods. SIAM, Philadelphia (2015)
Schumaker, L.L.: Solving elliptic PDE’s on domains with curved boundaries with an immersed penalized boundary method. J. Sci. Comp. 80(3), 1369–1394 (2019)
Schumaker, L.L., Yu, A.: Approximation by polynomial splines on curved triangulations. Comput. Aided Geom. Design 92, 102050 (2022)
Serbin, S.M.: Computational investigations of least-squares type methods for the approximate solution of boundary value problems. Math. Comp. 29, 777–793 (1975)
Wachspress, E.L.: A Rational Finite Element Basis. Academic Press, New York (2012)
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Schumaker, L.L. Solving Elliptic PDE’s Using Polynomial Splines on Curved Triangulations. J Sci Comput 92, 74 (2022). https://doi.org/10.1007/s10915-022-01932-6
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DOI: https://doi.org/10.1007/s10915-022-01932-6