1 /*! \file secOrd_op_solutionDriven.h
2 \brief Surface operator for the solution driven paper
3
4 Revision history
5 --------------------------------------------------
6
7 Revised by Christian Power April 2016
8 Originally written by Christian Power
9 (power22c@gmail.com) March 2016
10
11 Idea
12 --------------------------------------------------
13
14 Provides a class that perfoms the Dziuk mean curvature flow
15 scheme with an additional right-hand side.
16
17
18 \author Christian Power
19 \date 23. April 2016
20 \copyright Copyright (c) 2016 Christian Power. All rights reserved.
21 */
22
23 #ifndef SECORD_OP_SOLUTIONDRIVEN_H
24 #define SECORD_OP_SOLUTIONDRIVEN_H
25
26 #include <memory>
27 #include "esfem_fwd.h"
28
29 namespace Esfem{
30 namespace SecOrd_op{
31 class Solution_driven{
32 public:
33 //! Get PDE parameter, time and growth promoting function \f$u\f$.
34 /*! Extracts from `Io::Parameter` the parameter \f$ \alpha \f$
35 and \f$ \delta \f$.
36 `Grid::Grid_and_time` provides dynamical time steps via `time_provider`.
37 More precisely we need the method `velocity_regularization`,
38 `surface_growthFactor` and `mcf_regularization` from `Io::Parameter`.
39 \post Grid_and_time and `u_wrapper` outlive this object.
40 */
41 Solution_driven(const Io::Parameter&, const Grid::Grid_and_time&,
42 const Grid::Scal_FEfun& u_wrapper);
43 //! Required for the pointer to implementation technique.
44 ~Solution_driven();
45
46 //! Solve regularized mean curvature equation
47 /*! \param rhs \f$\nodalValue{Y}\f$
48 \retval lhs Solution of
49 \f$
50 \parentheses[\big]{M_3^n + (\alpha + \varepsilon\tau) A_3^n}
51 \nodalValue{X}^{n+1} = \nodalValue{Y}
52 \f$
53 \remark The solver is the conjugated gradient method.
54 \pre Parameter `rhs` must be assembled.
55 \sa brusselator_rhs(), Vec_rhs
56 */
57 void solve(const Grid::Vec_FEfun& rhs, Grid::Vec_FEfun& lhs) const;
58 //! Generates right-hand side without load vector for the linear system.
59 /*! \param rhs \f$\nodalValue{X}^n\f$
60 \retval lhs
61 \f$
62 (M_3^n + \alpha A_3^n) \nodalValue{X}^n
63 + \tau \delta M_3^n(\nodalValue{u}^n, \nodalValue{\surfaceNormal})
64 \f$
65 \pre \f$u\f$, which is given in the constructor, is still valid.
66 The nodal values of `rhs` should be the nodes itself.
67 \sa Identity
68 */
69 void brusselator_rhs(const Grid::Vec_FEfun& rhs, Grid::Vec_FEfun& lhs) const;
70 //! For debugging reasons
71 /*! \param rhs \f$\nodalValue{X}\f$
72 \retval lhs \f$ (M^n + (\alpha + \varepsilon \tau) A^n) \nodalValue{X} \f$
73 */
74 void operator()(const Grid::Vec_FEfun& rhs, Grid::Vec_FEfun& lhs) const;
75 private:
76 struct Data;
77 //! Pointer to data members
78 std::unique_ptr<Data> d_ptr;
79 };
80 /*!< \brief Solve the surface partial differential equation for
81 solution driven paper.
82
83 Partial differential equation
84 ==================================================
85
86 Parameter
87 --------------------------------------------------
88
89 \f$\varepsilon, \delta, a \in \R\f$ for the surface equation, where
90 \f$\varepsilon\f$ and \f$\alpha\f$ are small regularization parameter.
91
92 Smooth problem
93 --------------------------------------------------
94
95 For given \f$u\colon \surface \to \R\f$ search for
96 \f$X\colon \surface_{0} \times [0,T] \to \R^{m+1}\f$ such that
97 \f{align*}{
98 v - \alpha \laplaceBeltrami v = {} &
99 \parentheses[\big]{\varepsilon (-\meanCurvature) + \delta u} \surfaceNormal
100 = \varepsilon \laplaceBeltrami X + \delta u \surfaceNormal, \\
101 \dell_{t} X = {} & v(X).
102 \f}
103
104 Finite element discretization
105 --------------------------------------------------
106
107 For given \f$\nodalValue{u}\colon I \to \R^{N}\f$ search for
108 \f$\nodalValue{X}\colon I \to \R^{3N}\f$ (surface nodal values) such that
109 \f{equation*}{
110 \parentheses[\big]{M(X) + \alpha A(X)} \dell_t X =
111 \varepsilon A(X)X + \delta M(u,\nodalValue{\surfaceNormal})
112 \f}
113
114 Full discretization
115 --------------------------------------------------
116
117 For given \f$\nodalValue{X}^n\f$ and \f$\nodalValue{u}^n\f$ solve for
118 \f$\nodalValue{X}^{n+1}\f$
119 \f{equation*}{
120 \parentheses[\big]{M_3^n + (\alpha + \varepsilon\tau) A_3^n}
121 \nodalValue{X}^{n+1}
122 = (M_3^n + \alpha A_3^n) \nodalValue{X}^n
123 + \tau \delta M_3^n(\nodalValue{u}^n,
124 \nodalValue{\surfaceNormal}),
125 \f}
126 where \f$\nodalValue{\surfaceNormal}^n\f$ is elementwise normal.
127 */
128 }
129 } // Esfem
130
131 #endif // SECORD_OP_SOLUTIONDRIVEN_H