Reduced test spaces for DPG methods using rectangular elements

Abstract This paper shows that the test spaces in discontinuous Petrov Galerkin (DPG) methods can be reduced on rectangular elements without affecting unisolvency or rates of convergences. One reduced case is obtained by decreasing the polynomial degree of a standard test space in both coordinate directions by one. A further reduction of test space by almost another full degree is possible if one is willing to implement a nonstandard test space. The error analysis of such cases is based on an extension of the second Strang lemma and an interpretation of the DPG method as a nonconforming method. The key technical ingredient in obtaining unisolvency is the identification of a discontinuous piecewise polynomial on the element boundary that is orthogonal to all continuous piecewise polynomials of one degree higher.

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