Bivariate Spline Finite Element Solver for Linear Hyperbolic Equations in Two-Dimensional Spaces

In this paper, a finite element method using bivariate spline on domains is proposed for solving linear hyperbolic equations in two-dimensional spaces. Bivariate spline space $$S_{4}^{2,3} (\Delta_{mn}^{(2)} )$$S42,3(Δmn(2)) is constructed. It not only satisfies homogeneous boundary constraints but also satisfies interpolating boundary conditions on type-2 triangulations. Two examples are shown to confirm the correctness of theoretical conclusions. This means that spline method is efficient and feasible to solve 2D linear hyperbolic equations.

[1]  Ahmet Yildirim,et al.  He's homotopy perturbation method for solving the space- and time-fractional telegraph equations , 2010, Int. J. Comput. Math..

[2]  A. K. Pani,et al.  A posteriori error estimates for mixed finite element Galerkin approximations to second order linear hyperbolic equations , 2017 .

[3]  C. Provatidis.  B-splines collocation for plate bending eigenanalysis , 2017 .

[4]  Jieqing Tan,et al.  Generalized B-splines' geometric iterative fitting method with mutually different weights , 2018, J. Comput. Appl. Math..

[5]  A. Buffa,et al.  New refinable spaces and local approximation estimates for hierarchical splines , 2015, 1507.06534.

[6]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates , 2017 .

[7]  Jafar Biazar,et al.  Analytic solution for Telegraph equation by differential transform method , 2010 .

[8]  J. Rashidinia,et al.  Collocation method based on modified cubic B-spline for option pricing models , 2016 .

[9]  Andrea Barth,et al.  Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients , 2014 .

[10]  Alaattin Esen,et al.  Numerical Solution of Time Fractional Schrödinger Equation by Using Quadratic B-Spline Finite Elements , 2017 .

[11]  Hengfei Ding,et al.  A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation , 2009 .

[12]  Say Song Goh,et al.  From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa , 2014 .

[13]  Ren-Hong Wang,et al.  Bivariate quartic spline spaces and quasi-interpolation operators , 2006 .

[14]  Hendrik Speleers,et al.  Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis , 2017 .

[15]  Thomas J. R. Hughes,et al.  Isogeometric collocation for large deformation elasticity and frictional contact problems , 2015 .

[16]  Carlotta Giannelli,et al.  Adaptive isogeometric methods with hierarchical splines: error estimator and convergence , 2015, 1502.00565.

[17]  Eid H. Doha,et al.  New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations , 2016 .