| Summary: | In this work, we present the mathematical formulation of the new adaptive multiresolution method for the Stokes problems of highly viscous materials arising in computational geodynamics. The method is based on particle-in-cell approach—the Stokes system is solved on a static Eulerian finite element grid and material properties are carried in space by Lagrangian material points. The Eulerian grid is adapted using the wavelet-based adaptation algorithm. Both bilinear (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">P</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub><msub><mi mathvariant="normal">Q</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>) and biquadratic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">Q</mi><mn>2</mn></msub><msub><mi mathvariant="normal">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>) mixed approximations for the Stokes system are supported. The proposed method is illustrated for a number of linear and nonlinear two-dimensional benchmark problems of geophysical relevance. The results of the adaptive numerical simulations using the proposed method are in an excellent agreement with those obtained on non-adaptive grids and with analytical solutions, while computational requirements are few orders of magnitude less compared to the non-adaptive simulations in terms of both time and memory usage.
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