In this paper we study the convergence of monotone P1 finite element methods for fully nonlinear Hamilton–Jacobi–Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space L²(H¹γ(Ω)) to the viscosity solution without assuming uniform parabolicity of the HJB operator

Barucq, Hélène

Fares, M'Barek

Kruse, Carola

Tordeux, Sébastien

International audienceWe develop and analyze a high-order outgoing radiation boundary condition for solving three-dimensional scattering problems by elongated obstacles. This Dirichlet to Neumann (DtN) condition is constructed using the classical method of separation of variables that allows to define the scattered field in a truncated domain. It reads as an infinite series which is truncated for numerical purposes. The radiation condition is implemented in a finite element framework represented by a large dense matrix. Fortunately, the dense matrix can be decomposed into a full block matrix which involves the degrees of freedom on the exterior boundary and a sparse finite element matrix. The inversion of the full block is avoided by using a Sherman-Morrison algorithm which reduces the memory usage drastically. Despite being of high-order, this method has only a low memory cost. Many applications such as sonar, radar or geophysical exploration involve exterior Helmholtz problems which raise the question of handling the unbounded propagation domain. One accurate approach consists in using boundary element methods (BEM), see Bendali & Fares (2008) and their references. The linear partial differential equation is reformulated as an integral equation which is set on the boundary of the scatterer. BEMs have thus the advantage of providing an exact reformulation of the exterior problem in a bounded region without introducing any kind of approximation before discretization. However , boundary element discretizations lead to dense matrices which increase both storage requirements and computational costs of the solution algorithm, especially in three dimensions. A popular alternative is the usage of volume finite element methods (FEM) Harari & Hughes (1992b) once the computational domain has been truncated. The truncation is performed by introducing an artificial boundary surrounding the scatterer which requires transforming the exterior problem into a mixed one corresponding to the initial volume formulation of the problem set in the bounded domain and coupled with a suitable boundary condition set on the external boundary. To get an accurate representation of the wave field into the bounded domain, the external boundary should not generate any reflection. This point is managed with 

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Sparsified discrete wave problem involving radiation condition on a prolate spheroidal surface

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In this paper we study the convergence of monotone P1

 finite element methods for fully nonlinear Hamilton–Jacobi–Bellman (HJB) equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space L2(H1γ(Ω))

 to the viscosity solution without assuming uniform parabolicity of the HJB operator

Jensen, Max

UCL Discovery

English

L2(H1γ) Finite Element Convergence for Degenerate Isotropic Hamilton–Jacobi–Bellman Equations

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https://hal.inria.fr/hal-02386957/file/main.pdf

L²(H¹γ) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations

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HAL-Rennes 1

INRIA a CCSD electronic archive server

UCL Discovery

Archive Ouverte en Sciences de l'Information et de la Communication