1 /*! \file secOrd_op_linearHeat.h
2 \author Christian Power
3 \date 17. March 2016
4
5 \brief Standard heat resp. diffusion equation for an
6 evolving or stationary surface problem
7
8 Revision history
9 --------------------------------------------------
10
11 Revised by Christian Power dd.mm.yyyy
12 Originally written by Christian Power
13 (power22c@gmail.com) 28. Januar 2016
14
15 Idea
16 --------------------------------------------------
17
18 Provides a class that performs the standard Dziuk Elliott evolving surface
19 finite element discretization with implicit euler time discretization.
20
21 Partial differential equation
22 ==================================================
23
24 Parameter
25 --------------------------------------------------
26
27 - Right-hand side function f
28 - Function u_0 for start time t_0
29 - Surface velocity v
30
31 Smooth problem
32 --------------------------------------------------
33
34 Search for \f$u\colon \surface \to \R\f$ for
35 \f{equation*}{
36 \matd u + u \diver(v) - \laplaceBeltrami u = f
37 \f}
38
39 Finite element discretization
40 --------------------------------------------------
41
42 Search for \f$\nodalValue{u}\colon I \to \R^N \f$ for
43 \f{equation*}{
44 \dell_t \parentheses[\big]{M(t) \nodalValue{u} } + A(t) \nodalValue{u}
45 = M(t)\nodalValue{Pf},
46 \f}
47 where \f$ \nodalValue{Pf} \f$ are the nodal values of the
48 \f$ L^2 \f$-projection of \f$ f \f$.
49
50 Full discretization
51 --------------------------------------------------
52
53 Given \f$ \nodalValue{u}^n \f$ solve for \f$ \nodalValue{u}^{n+1} \f$
54 \f{equation*}{
55 (M\nodalValue{u})^{n+1} + \tau (A \nodalValue{u})^{n+1}
56 = (M \nodalValue{w})^n + \tau (M \nodalValue{Pf})^{n+1}
57 \f}
58
59 This programm implements a basic expression calculator.
60 Input from cin; output to cout. The grammar for input is: etc.
61
62 Created by Christian Power on 28.01.2016
63 Copyright (c) 2016 Christian Power. All rights reserved.
64 */
65
66 #ifndef SECORD_OP_LINEARHEAT_H
67 #define SECORD_OP_LINEARHEAT_H
68
69 #include <memory>
70 #include "esfem_fwd.h"
71
72 namespace Esfem{
73 namespace SecOrd_op{
74 class Linear_heat{
75 public:
76 explicit Linear_heat(const Io::Parameter&, const Grid::Grid_and_time&);
77 ~Linear_heat();
78
79 void solve(const Grid::Scal_FEfun& rhs, Grid::Scal_FEfun& lhs) const;
80 void apply_massMatrix_to(Grid::Scal_FEfun&) const;
81 void mass_matrix(const Grid::Scal_FEfun& rhs, Grid::Scal_FEfun& lhs) const;
82 private:
83 struct Data;
84 std::unique_ptr<Data> d_ptr;
85 };
86 }
87 }
88
89 #endif // SECORD_OP_LINEARHEAT_H