Numerische Integration über zweidimensionalen Gebieten
نویسنده
چکیده
This thesis discusses strategies for numerically solving a two dimensional integral. The difficulties here are due to numerous integrands and regions in R2. The transformation to units as the 2-simplex and the cube can simplify many integrands. An approach to arbitrary regions is to decompose them into easy ones as triangles and rectangles. For these regions formulas that approximate the two dimensional integral are developed. Newton-Cotes-Formulas use, like in one dimension, an interpolation of the integrand on fixed, equidistant nodes. Gauss-Formulas reach a higher precision than the Newton-Cotes-Formulas with free nodes. To find formulas with a minimum number of nodes, one analyses orthogonal polynomials and their common zeros. Although the zeros of orthogonal polynomials are used as nodes for formulas of maximum degree in one dimension, they can only be taken as nodes for two dimensional formulas under certain restrictions. Moreover the theory of orthogonal polynomials is much more complicated in more than one dimension. In the end, the derived formulas are demonstrated on some examples. An adaptive algorithm which divides a triangular region until it reaches a presigned tolerance is presented.
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