Outline The Mimetic Finite Difference Method Gianmarco Manzini 1 Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI) C.N.R., Pavia, Italy FVCA5 - June 08-13, 2008 Aussois, France Manzini, G. The Mimetic Finite Difference Method
Outline Outline MFD method for Darcy’s problem 1 Formal construction Mimetic conservation equation Mimetic constitutive equation Theoretical results and applications 2 A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation 3 Summary Manzini, G. The Mimetic Finite Difference Method
Outline Outline MFD method for Darcy’s problem 1 Formal construction Mimetic conservation equation Mimetic constitutive equation Theoretical results and applications 2 A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation 3 Summary Manzini, G. The Mimetic Finite Difference Method
Outline Outline MFD method for Darcy’s problem 1 Formal construction Mimetic conservation equation Mimetic constitutive equation Theoretical results and applications 2 A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation 3 Summary Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation “Mimetic (mathematics)” From Wikipedia, the free encyclopedia ( i ) The goal of numerical analysis is to approximate the continuum, so instead of solving a partial differential equation one aims in solve a discrete version of the continuum problem. ( ii ) A numerical method is called mimetic when it mimics (or imitates) some properties of the continuum vector calculus. An example: a mixed finite element method applied to Darcy flows strictly conserves the mass of the flowing fluid. Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Some literature. . . Mimetic schemes were first proposed in the early eighties: Samarskii-Tishkin-Favorskii-Shashkov, Operational Finite-Difference Schemes , Differential Equations, 1981; many papers were published after this one. . . Some recent joint work from Los Alamos-Pavia: Brezzi-Lipnikov-Shashkov,SINUM,2005 ( a priori estimates) Brezzi-Lipnikov-Simoncini, M 3 AS, 2005 (a family of MFDs) . . . Extensions: Cangiani-M. CMAME, 2008 (post-processing) Beirao da Veiga-M., NME, 2008 (mesh adaptivity) . . . Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation . . . people and topics (in Pavia) Main features: family of schemes based on mixed formulation; grids formed by elements of general shape (polygons, polyhedra); people currently working in Pavia: Beirao da Veiga, Boffi, Brezzi, Buffa, Cangiani, M., A. Russo, . . . some topics under investigation: diffusion and convection-diffusion models a posteriori estimates and mesh adaptivity electromagnetism Stokes equations 2-D software implementation Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Outline MFD method for Darcy’s problem 1 Formal construction Mimetic conservation equation Mimetic constitutive equation Theoretical results and applications 2 A priori estimates Post-processing of solution An error estimator and mesh adaptivity Convection-Diffusion-Reaction Equation 3 Summary Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Linear diffusion in mixed form Consider div ( − K ∇ p ) = b , Ω ⊂ I R d , d = 2 , 3 +boundary conditions Let − → F be the flux vector variable: − → F = − K ∇ p constitutive equation (1) div − → F = b conservation equation Model problem: solve (1) for p and − → F with suitable boundary conditions Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Formally FIRST , (let {T h } be a family of partitions of Ω formed by polygonal elements, h being the mesh size); ( i ) degrees of freedom for − scalar fields − → discrete scalars , Q h ; − vector fields − → discrete vectors , X h ; Q h and X h are not functions, but vectors of numbers! ( ii ) “discrete” operators: − the discrete divergence DIV h : X h → Q h ; − the discrete flux (or gradient) G h : Q h → X h ; satisfying a duality relationship (discrete Gauss-Green formula). Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Formally FIRST , (let {T h } be a family of partitions of Ω formed by polygonal elements, h being the mesh size); ( i ) degrees of freedom for − scalar fields − → discrete scalars , Q h ; − vector fields − → discrete vectors , X h ; Q h and X h are not functions, but vectors of numbers! ( ii ) “discrete” operators: − the discrete divergence DIV h : X h → Q h ; − the discrete flux (or gradient) G h : Q h → X h ; satisfying a duality relationship (discrete Gauss-Green formula). Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Formally FIRST , (let {T h } be a family of partitions of Ω formed by polygonal elements, h being the mesh size); ( i ) degrees of freedom for − scalar fields − → discrete scalars , Q h ; − vector fields − → discrete vectors , X h ; Q h and X h are not functions, but vectors of numbers! ( ii ) “discrete” operators: − the discrete divergence DIV h : X h → Q h ; − the discrete flux (or gradient) G h : Q h → X h ; satisfying a duality relationship (discrete Gauss-Green formula). Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Formally THEN , mimic the continuous differential equations by using discrete operators acting on the discrete scalar and flux unknowns p h ∈ Q h and F h ∈ X h : constitutive equation: − → F = − K ∇ p − → F h = G h p h conservation equation: div − → DIV h F h = b I F = b − → (where b I is a suitable interpolation of b in Q h ) Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Q h , degrees of freedom for scalar fields ˘ ¯ • q ∈ Q h means q = q E Ω , T h E ∈T h (equivalent to a piecewise constant function) • dim ( Q h ) = number of elements E q E of the mesh. • ”interpolation” operator: 1 Z ( p I ) E = p dV . | E | E Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Q h , degrees of freedom for scalar fields ˘ ¯ • q ∈ Q h means q = q E Ω , T h E ∈T h (equivalent to a piecewise constant function) • dim ( Q h ) = number of elements E q E of the mesh. • ”interpolation” operator: 1 Z ( p I ) E = p dV . | E | E Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation Q h , degrees of freedom for scalar fields ˘ ¯ • q ∈ Q h means q = q E Ω , T h E ∈T h (equivalent to a piecewise constant function) • dim ( Q h ) = number of elements E q E of the mesh. • ”interpolation” operator: 1 Z ( p I ) E = p dV . | E | E Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation X h , degrees of freedom for vector fields G e ˘ ¯ • G ∈ X h means G = E Ω , T h e is an edge of E G e E + G e ∀ e ⊆ E ∩ E ′ • E ′ = 0 dim ( X h ) = number of edges e of the mesh. E • ”interpolation” operator: ` − → E · − → E = 1 Z − → F I ´ e E ′ n e F dV | e | e − → n e E E Manzini, G. The Mimetic Finite Difference Method
Mimetic Formulation Formal construction Theoretical results and applications Mimetic conservation equation Summary Mimetic constitutive equation X h , degrees of freedom for vector fields G e ˘ ¯ • G ∈ X h means G = E Ω , T h e is an edge of E G e E + G e ∀ e ⊆ E ∩ E ′ • E ′ = 0 dim ( X h ) = number of edges e of the mesh. E • ”interpolation” operator: ` − → E · − → E = 1 Z − → F I ´ e E ′ n e F dV | e | e − → n e E E Manzini, G. The Mimetic Finite Difference Method
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