Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons
نویسندگان
چکیده
We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre’s method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.
منابع مشابه
Bernstein's polynomials for convex functions and related results
In this paper we establish several polynomials similar to Bernstein's polynomials and several refinements of Hermite-Hadamard inequality for convex functions.
متن کاملNumerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra
We present a method for the numerical integration of homogeneous functions over convex and nonconvex polygons and polyhedra. On applying Stokes’s theorem and using the property of homogeneous functions, we show that it suffices to integrate these functions on the boundary facets of the polytope. For homogeneous polynomials, this approach is used to further reduce the integration to just functio...
متن کاملQuadratic maximum-entropy serendipity shape functions for arbitrary planar polygons
In this paper, we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons. Drawing on the work of Bompadre et al. [1] and Hormann and Sukumar [2], we adopt a relative entropy measure for signed (positive or negative) shape functions, with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon. We maximize t...
متن کاملNumerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials
In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...
متن کاملNumerical Solution of Weakly Singular Ito-Volterra Integral Equations via Operational Matrix Method based on Euler Polynomials
Introduction Many problems which appear in different sciences such as physics, engineering, biology, applied mathematics and different branches can be modeled by using deterministic integral equations. Weakly singular integral equation is one of the principle type of integral equations which was introduced by Abel for the first time. These problems are often dependent on a noise source which a...
متن کامل