Discrete conservation of nonnegativity for elliptic problems solved by the \(hp\)-FEM. (English) Zbl 1135.65395
Summary: Most results related to discrete nonnegativity conservation principles (DNCP) for elliptic problems are limited to finite differences and lowest-order finite element methods (FEM). In this paper we show that a straightforward extension to higher-order finite element methods (\(hp\)-FEM) in the classical sense is not possible. We formulate a weaker DNCP for the Poisson equation in one spatial dimension and prove it using an interval computing technique. Numerical experiments related to the extension of this result to 2D are presented.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
discrete nonnegativity conservation; discrete Green’s function; elliptic problems; \(hp\)-FEM; higher-order finite element methods; Poisson equation; numerical experimentsReferences:
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