Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

Abstract We present quadrature schemes to calculate matrices where the so-called modified Hilbert transformation is involved. These occur as temporal parts of Galerkin finite element discretizations parabolic or hyperbolic problems when used for variational setting. This work provides calculation these machine precision arbitrary polynomial degrees and non-uniform meshes. The proposed are based on weakly singular integral representations transformation. First, proven. Second, using representations, we derive schemes, which treat occurring singularities appropriately. Thus, exponential convergence with respect number nodes achieved. Numerical results, this observed, conclude work.