22,231 research outputs found
Convergence of adaptive discontinuous Galerkin and Cā°-interior penalty finite element methods for HamiltonāJacobiāBellman and Isaacs equations
We prove the convergence of adaptive discontinuous Galerkin and C0-interior penalty methods for fully nonlinear second-order elliptic HamiltonāJacobiāBellman and Isaacs equations with Cordes coefficients. We consider a broad family of methods on adaptively refined conforming simplicial meshes in two and three space dimensions, with fixed but arbitrary polynomial degrees greater than or equal to two. A key ingredient of our approach is a novel intrinsic characterization of the limit space that enables us to identify the weak limits of bounded sequences of nonconforming finite element functions. We provide a detailed theory for the limit space, and also some original auxiliary functions spaces, that is of independent interest to adaptive nonconforming methods for more general problems, including PoincarĆ© and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, approximation results by finite element functions and weak convergence results
Convergence of adaptive discontinuous Galerkin and -interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations
We prove the convergence of adaptive discontinuous Galerkin and
-interior penalty methods for fully nonlinear second-order elliptic
Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We
consider a broad family of methods on adaptively refined conforming simplicial
meshes in two and three space dimensions, with fixed but arbitrary polynomial
degrees greater than or equal to two. A key ingredient of our approach is a
novel intrinsic characterization of the limit space that enables us to identify
the weak limits of bounded sequences of nonconforming finite element functions.
We provide a detailed theory for the limit space, and also some original
auxiliary functions spaces, that is of independent interest to adaptive
nonconforming methods for more general problems, including Poincar\'e and trace
inequalities, a proof of density of functions with nonvanishing jumps on only
finitely many faces of the limit skeleton, approximation results by finite
element functions and weak convergence results
A finite element method for fully nonlinear elliptic problems
We present a continuous finite element method for some examples of fully
nonlinear elliptic equation. A key tool is the discretisation proposed in
Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of
a linear PDE. An added benefit to making use of this discretisation method is
that a recovered (finite element) Hessian is a biproduct of the solution
process. We build on the linear basis and ultimately construct two different
methodologies for the solution of second order fully nonlinear PDEs. Benchmark
numerical results illustrate the convergence properties of the scheme for some
test problems including the Monge-Amp\`ere equation and Pucci's equation.Comment: 22 pages, 31 figure
Applications of nonvariational finite element methods to Monge--Amp\`ere type equations
The goal of this work is to illustrate the application of the nonvariational
finite element method to a specific Monge--Amp\`ere type nonlinear partial
differential equation. The equation we consider is that of prescribed Gauss
curvature.Comment: 7 pages, 3 figures, tech repor
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